Quadratic vector equations on complex upper half-plane
QQuadratic vector equations oncomplex upper half-plane
Oskari H. Ajanki ∗ IST Austria [email protected]
László Erdős † IST Austria [email protected]
Torben Krüger ‡ IST Austria [email protected]
Abstract
We consider the nonlinear equation − m = z + Sm with a parameter z in the complex upper half plane H , where S is a positivity preservingsymmetric linear operator acting on bounded functions. The solution withvalues in H is unique and its z -dependence is conveniently described as theStieltjes transforms of a family of measures v on R . In [ ? ] we qualitativelyidentified the possible singular behaviors of v : under suitable conditionson S we showed that in the density of v only algebraic singularities ofdegree two or three may occur. In this paper we give a comprehensiveanalysis of these singularities with uniform quantitative controls. We alsofind a universal shape describing the transition regime between the squareroot and cubic root singularities. Finally, motivated by random matrixapplications in the companion paper [AEK16c], we present a completestability analysis of the equation for any z ∈ H , including the vicinity ofthe singularities. Keywords:
Stieltjes-transform, Algebraic singularity, Density of states,Cubic cusp, Wigner-type random matrix.
AMS Subject Classification (2010): , , , . ∗ Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12Grant of the German Research Council. † Partially supported by ERC Advanced Grant RANMAT No. 338804. ‡ Partially supported by ERC Advanced Grant RANMAT No. 338804, and SFB-TR 12Grant of the German Research Council a r X i v : . [ m a t h . P R ] A ug ontents AEK16b ] 182.4 Outline of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 L -bound 31 F and structural L -bound . . . . . . . . . . . . . . . . 37 m and F . . . . . . . . . . . . 395.2 Stability and operator B . . . . . . . . . . . . . . . . . . . . . . . 47 L -estimates . . . . . . . . . . . . . . . . . 566.2 Uniform bound around z = 0 when a = 0 . . . . . . . . . . . . . 58 B . . . . . . . . . . . . . . . . . . . . . . . 718.2 Cubic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A3 . . . 12311.2 Divergences in B , outliers, and function Γ . . . . . . . . . . . . . 12411.2.1 Simplest example of blow-up in B : . . . . . . . . . . . . . 12411.2.2 Example of blow-up in B due to smoothing: . . . . . . . 12611.3 Blow-up at z = 0 when a = 0 and assumption B1 . . . . . . . . . 12711.4 Effects of non-constant function a . . . . . . . . . . . . . . . . . . 12911.5 Discretization and reduction of the QVE . . . . . . . . . . . . . . 12911.6 Simple example that exhibits all universal shapes . . . . . . . . . 131 A Appendix 133
A.1 Proofs of auxiliary results in Chapter 4 . . . . . . . . . . . . . . . 135A.2 Proofs of auxiliary results in Chapter 5 . . . . . . . . . . . . . . . 135A.3 Scalability of matrices with non-negative entries . . . . . . . . . . 137A.4 Variational bounds when Re z = 0 . . . . . . . . . . . . . . . . . 141A.5 Hölder continuity of Stieltjes transform . . . . . . . . . . . . . . . 144A.6 Cubic roots and associated auxiliary functions . . . . . . . . . . . 146iii hapter 1 Introduction
One of the basic problems in the theory of large random matrices is to computethe asymptotic density of eigenvalues as the dimension of the matrices goes toinfinity. For several prominent ensembles this question is ultimately related tothe solution of a system of nonlinear equations of the form − m i = z + a i + N (cid:88) j =1 s ij m j , i = 1 , . . . , N , (1.1)where the complex parameter z and the unknowns m , . . . , m N lie in the complexupper half-plane H := { z ∈ C : Im z > } . The given vector a = ( a i ) Ni =1 hasreal components, and the matrix S = ( s ij ) Ni,j =1 is symmetric with non-negativeentries and it is determined by the second moments of the matrix ensemble.The simplest example for the emergence of (1.1) are the Wigner-type matri-ces , defined as follows. Let H = ( h ij ) be an N × N real symmetric or complexhermitian matrix with expectations E h ij = − a i δ ij and variances E | h ij | = s ij .We assume that the matrix elements are independent up to the symmetry con-straint, h ji = h ij . Let G ( z ) = ( H − z ) − be the resolvent of H with a spectralparameter z ∈ H . Second order perturbation theory indicates that for the diag-onal matrix elements G ii = G ii ( z ) of the resolvent we have − G ii ≈ z + a i + N (cid:88) j =1 s ij G jj , (1.2)where the error is due to fluctuations that vanish in the large N limit. Inparticular, if the system of equations (1.1) is stable, then G ii is close to m i and the average N − (cid:80) i m i approximates the normalized trace of the resol-vent, N − Tr G . Being determined by N − Im Tr G , as Im z → , the empiricalspectral measure of H approaches the non-random measure with density ρ ( τ ) := lim η ↓ πN N (cid:88) j =1 Im m j ( τ + i η ) , τ ∈ R , (1.3) 1s N goes to infinity, see [Shl96, Gui02, AZ05]. Apart from a few specific cases,this procedure via (1.1) is the only known method to determine the limitingdensity of eigenvalues for large Wigner-type random matrices.When S is doubly stochastic, i.e., (cid:80) j s ij = 1 for each row i , then it is easyto see that the only solution to (1.1) is the constant vector, m i = m sc for each i , where m sc = m sc ( z ) is the Stieltjes transform of Wigner’s semicircle law, m sc ( z ) = (cid:90) R ρ sc ( τ )d ττ − z , with ρ sc ( τ ) := 12 π (cid:112) max { , − τ } . (1.4)The system of equations (1.1) thus reduces to the simple scalar equation − m sc = z + m sc . (1.5)Comparing (1.2) and (1.1), we see from (1.3) that the density of the eigenvaluesin the large N limit is given by the semicircle law. The corresponding randommatrix ensemble was called generalized Wigner ensemble in [EYY11a].Besides Wigner-type matrices and certain random matrices with translationinvariant dependence structure [AEK16a], the equation (1.1) has previouslyappeared in at least two different contexts. First, in [AZ08] the limiting densityof eigenvalues for a certain class of random matrix models with dependent entrieswas determined by the so-called color equations (cf. equation (3.9) in [AZ08]),which can be rewritten in the form (1.1). For more details on this connection werefer to Subsection 3.4 of [AEK16b]. The second application of (1.1) concernsthe Laplace-like operator, ( Hf )( x ) = (cid:88) y ∼ x t xy ( f ( x ) − f ( y )) , f : V → C on rooted tree graphs Γ with vertex set V (see [KLW] for a review article andreferences therein). Set m x = ( H x − z ) − ( x, x ) , where H x is the operator H restricted to the forward subtree with root x . A simple resolvent formula thenshows that (1.1) holds with a = 0 and s xy = | t xy | { x < y } , where x < y indicates that x is closer to the root of Γ than y . In this example ( s xy ) is not asymmetric matrix, but in a related model it may be chosen symmetric (rootedtrees of finite cone types associated with a substitution matrix S , see [Sad12]).In particular, real analyticity of the density of states (away from the spectraledges) in this model follows from our analysis (We thank C. Sadel for pointingout this connection).The central role of (1.1) in the context of random matrices has been rec-ognized by many authors, see, e.g. [Ber73, Weg79, Gir01, KP94, Shl96, AZ08,Gui02] and some basic properties of the solution, such as existence, unique-ness and regularity in z away from the real axis have been established, see e.g.[Gir01, HFS07, PS11] and further references therein. The existence of the limitin (1.3) has been shown but no description of the limiting density ρ was given.2otivated by this problem, in [AEK16b] we initiated a comprehensive studyof a general class of nonlinear equations of the form − m = z + a + Sm , in a possibly infinite dimensional setup. Under suitable conditions on the linearoperator S , we gave a qualitative description of the possible singularities of m as z approaches the real axis. We showed that singularities can occur at mostat finitely many points and that they are algebraic of order two or three. Thesolution m is conveniently represented as the Stieltjes transforms of a family ofprobability measures. The singularities of m occur at points where the densitiesof these measures approach to zero and the type of singularity depends on howthe densities vanish. We found that the densities behave like a square root nearthe edges of their support and, additionally, they may exhibit a cubic root cuspsingularity inside the interior of the support; no other singularity type occurs.All these results translate into statements about the spectral densities oflarge random matrices on the macroscopic scale. Recent developments in thetheory of random matrices, however, focus on local laws , i.e. precise descriptionof the eigenvalue density down to very small scales almost comparable with theeigenvalue spacing. This requires understanding the solution m and the stabilityof (1.1) with an effective quantitative control as z approaches the real line. Inparticular, a detailed description of the singular behavior of the solution closeto the spectral edges is necessary.The current paper is an extensive generalization of the qualitative singular-ity analysis of [AEK16b]. Here we give a precise description (cf. Theorem 2.6below) of the density around the singularities in a neighborhood of order onewith effective error bounds, while in [AEK16b] we only proved the limiting be-havior as z approached the singularities without uniform control. We analyzethe density around all local minima inside the interior of the support, even whenthe value of the density is small but non-zero. We demonstrate that a universal density shape emerges also at these points, which are far away from any singular-ity. In Subsection 2.3 we demonstrate the strength of the current bounds overthe qualitative results in [AEK16b] by considering a one-parameter family ofoperators S . By varying the parameter, this family exhibits all possible shapesof the density in the regime where the density is close to zero. This exampleillustrates how these density shapes are realized as rescalings of two universalshape functions. Furthermore, in the current work we impose weaker conditionson a and S than in [AEK16b]. Especially, when a = 0 our assumptions on S are essentially optimal. Finally, we also give a detailed stability analysis againstsmall perturbations; the structure of our stability bounds is directly motivatedby their application in random matrices.The uniform control of the solution near the singularities, as well as thequantitative stability estimates are not only natural mathematical questions ontheir own. They are also indispensable for establishing local laws and univer-sality of local spectral statistics of Wigner type random matrices. We heavilyuse them in the companion papers to prove such results for Wigner-type matri-3es with independent entries [AEK16c], as well as for matrices with correlatedGaussian entries [AEK16a]. Random matrices, however, will not appear in themain body of this work. In Chapter 3 we only illustrate how our analysis of(1.1) is used to prove a simple version of the local law.While the current work is a generalization of [AEK16b], it is essentially self-contained; only very few auxiliary results will be taken over from [AEK16b].To achieve the required uniform control, we need to restart the analysis fromits beginning. After establishing a priori bounds on the solution m and onthe stability of the linearization of (1.1) in Chapters 4–6, there are two mainsteps. First, in Chapter 8 we derive an approximate cubic equation to determinethe leading behavior of the density in the regime where it is very small and,second, we analyze this cubic equation. The first step is much more involvedin this paper than in [AEK16b] since we also need to analyze points where thedensity is small but nonzero and we require all bounds to be effective in termsof a small number of model parameters. The second step follows a completelynew argument. In [AEK16b] the correct roots of the cubic equation have beenselected locally and by using a proof by contradiction which cannot give anyeffective control. In the current paper we select the roots by matching thesolutions at neighboring singularities to ensure the effective control in an orderone neighborhood. This procedure takes up Chapter 9, the most technical partof our work. The nonlinear stability analysis is presented in Chapter 10; thisis the strongest version needed in the random matrix analysis in [AEK16c].Finally, in Chapter 11 we present examples illustrating various aspects of themain results and the necessity of the assumptions on a and S . Acknowledgement.
We are grateful to Zhigang Bao and Christian Sadel forseveral comments and suggestions along this work.4 hapter 2
Set-up and main results
In this chapter we formulate a generalized version of the equation (1.1) whichallows us to treat all dimensions N , including the limit N → ∞ , in a unifiedmanner. After introducing three assumptions A1-3 on a and S we state ourmain results.Let X be an abstract set of labels. We introduce the Banach space, B := (cid:110) w : X → C : sup x ∈ X | w x | < ∞ (cid:111) , (2.1)of bounded complex valued functions on X , equipped with the norm (cid:107) w (cid:107) := sup x ∈ X | w x | . (2.2)We also define the subset B + := (cid:110) w ∈ B : Im w x > for all x ∈ X (cid:111) , (2.3)of functions with values in the complex upper half-plane H .Let S : B → B be a non-zero bounded linear operator, and a ∈ B areal valued bounded function. The main object of study in this paper is theequation, − m ( z ) = z + a + S m ( z ) , ∀ z ∈ H , (2.4)and its solution m : H → B + . Here we view m : X × H → H , ( x, z ) (cid:55)→ m x ( z ) asa function of the two variables x and z , but we will often suppress the x and/or z dependence of m and other related functions. The symbol m will always refer toa solution of (2.4). We will refer to (2.4) as the Quadratic Vector Equation ( QVE ).We assume that X is equipped with a probability measure π and a σ -algebra S such that ( X , S , π ) constitutes a probability space. We will denote the space5f measurable functions u : X → C , satisfying (cid:107) u (cid:107) p := ( (cid:82) X | u x | p π (d x )) /p < ∞ ,as L p = L p ( X ; C ) , p ≥ . The usual L -inner product, and the averaging aredenoted by (cid:104) u, w (cid:105) := (cid:90) X u x w x π (d x ) , and (cid:104) w (cid:105) := (cid:104) , w (cid:105) , u, w ∈ L , (2.5)respectively. For a linear operator A , mapping a Banach space X to anotherBanach space Y , we denote the corresponding operator norm by (cid:107) A (cid:107) X → Y . How-ever, when X = Y = B we use the shorthand (cid:107) A (cid:107) = (cid:107) A (cid:107) B → B . Finally, if w isa function on X and T is a linear operator acting on such functions then w + T denotes the linear operator u (cid:55)→ wu + T u , i.e., we interpret w as a multiplicationoperator when appropriate.In the entire paper we assume that the bounded linear operator S : B → B in (2.4) is: A1 Symmetric and positivity preserving , i.e., for every u, w ∈ B and everyreal valued and non-negative p ∈ B : (cid:104) u, Sw (cid:105) = (cid:104) Su, w (cid:105) , and inf x ( Sp ) x ≥ . For the existence and uniqueness no other assumptions on a and S areneeded. Theorem 2.1 (Existence and uniqueness) . Assume A1 . Then for each z ∈ H , − m = z + a + S m , (2.6) has a unique solution m = m ( z ) ∈ B + . The solutions for different values of z constitute an analytic function z (cid:55)→ m ( z ) from H to B + . Moreover, for each x ∈ X there exists a positive measure v x on R , with supp v x ⊂ [ − Σ , Σ ] , where Σ := (cid:107) a (cid:107) + 2 (cid:107) S (cid:107) / , (2.7) and v x ( R ) = π , such that m x ( z ) = 1 π (cid:90) R v x (d τ ) τ − z , ∀ z ∈ H . (2.8) The measures v x constitute a measurable function v := ( x (cid:55)→ v x ) : X → M ( R ) ,where M ( R ) denotes the space of finite Borel measures on R equipped with theweak topology.Furthermore, if a = 0 , then the solution m ( z ) is in L whenever z (cid:54) = 0 , (cid:107) m ( z ) (cid:107) ≤ | z | , ∀ z ∈ H , (2.9) and the measures v x are symmetric, in the sense that v x ( − A ) = v x ( A ) for anymeasurable set A ⊂ R . L -bound (2.9) for thecase a = 0 , which is proven in Chapter 5.We remark that if the solution space B + is replaced by B , then the equation(2.6) in general may have multiple, even infinitely many, solutions. Since v gen-erates the solution m through (2.8) we call the x -dependent family of measures v = ( v x ) x ∈ X the generating measure .In order to prove results beyond the existence and uniqueness we need toadditionally assume that S is: A2 Smoothing , in the sense that it extends to a bounded operator from L to B that is represented by a symmetric non-negative measurable kernelfunction ( x, y ) (cid:55)→ S xy : X → [0 , ∞ ) , i.e., (cid:107) S (cid:107) L → B < ∞ , and ( Sw ) x = (cid:90) X S xy w y π (d y ) . (2.10) A3 Uniformly primitive , i.e., there exist an integer L ∈ N , and a constant ρ > , such that u ∈ B , u ≥ ⇒ ( S L u ) x ≥ ρ (cid:104) u (cid:105) ∀ x ∈ X . (2.11)The finiteness of the norm (cid:107) S (cid:107) L → B in condition A2 means that the integralkernel S xy representing the operator S satisfies (cid:107) S (cid:107) L → B = sup x ∈ X (cid:16) (cid:90) X ( S xy ) π (d y ) (cid:17) / < ∞ . (2.12)In particular, S is a Hilbert-Schmidt operator on L . The condition A3 is aneffective lower bound on the coupling between the components m x in the QVE.In the context of matrices with non-negative entries this property is known as primitivity - hence our terminology. Remark . If we replace the pair ( a, S ) in the QVEwith ( a (cid:48) , S (cid:48) ) := ( λ / a + τ , λS ) , for some constants λ > and τ ∈ R , then themodified QVE is solved by m (cid:48) : H → B , where m (cid:48) x ( z ) := λ − / m x ( λ − / ( z − τ )) .By this basic observation, we may assume, without loss of generality, that S isnormalized and a is centered, i.e., (cid:107) S (cid:107) = 1 and (cid:104) a (cid:105) = 0 .All important estimates in this paper are quantitative in the sense that theydepend on a and S only through a few special parameters (see also Section 2.3).The following convention makes keeping track of this dependence easier. Convention 2.3 (Comparison relations, model parameters and constants) . Forbrevity we introduce the concept of comparison relations : If ϕ = ϕ ( u ) and7 = ψ ( u ) are non-negative functions on some set U , then the notation ϕ (cid:46) ψ ,or equivalently, ψ (cid:38) ϕ , means that there exists a constant < C < ∞ such that ϕ ( u ) ≤ C ψ ( u ) for all u ∈ U . If ψ (cid:46) ϕ (cid:46) ψ then we write ϕ ∼ ψ , and say that ϕ and ψ are comparable . Furthermore, we use ψ = φ + O X ( ξ ) as a shorthandfor (cid:107) ψ − φ (cid:107) X (cid:46) (cid:107) ξ (cid:107) X , where ξ, ψ, ϕ ∈ X and X is a normed vector space.For X = C we simply write O instead of O C . When the implicit constants C in the comparison relations depend on some parameters Λ we say that the comparison relations depend on Λ . Typically, Λ contains the parametersappearing in the hypotheses, and we refer to them as model parameters .We denote by C, C (cid:48) , C , C , . . . and c, c (cid:48) , c , c , . . . , etc., generic constantsthat depend only on the model parameters. The constants C, C (cid:48) , c, c (cid:48) maychange their values from one line to another, while the enumerated constants,such as c , C , have constant values within an argument or a proof.We usually express the dependence on the variable z explicitly in the state-ments of theorems, etc. However, in order to avoid excess clutter we oftensuppress the variable z within the proofs e.g., we write m instead of m ( z ) , when z is considered fixed. This section contains our main results, Theorem 2.4 and Theorem 2.6, con-cerning the generating measure, when S satisfies A1-3 , and the solution of theQVE is uniformly bounded. Sufficient conditions on S and a that guarantee theuniformly boundedness of m are also given (cf. Theorem 2.10).For any I ⊆ R we introduce the seminorm on functions w : H → B : ||| w ||| I := sup (cid:8) (cid:107) w ( z ) (cid:107) : Re z ∈ I , Im z ∈ (0 , ∞ ) (cid:9) . (2.13) Theorem 2.4 (Regularity of generating density) . Suppose S satisfies A1-3 ,and the solution m of (2.4) is uniformly bounded everywhere, i.e., ||| m ||| R ≤ Φ , for some constant Φ < ∞ . Then the following hold true: (i) The generating measure has a Lebesgue density (also denoted by v ), i.e., v x (d τ ) = v x ( τ )d τ . The components of the generating density are com-parable, i.e., v x ( τ ) ∼ v y ( τ ) , ∀ τ ∈ R , ∀ x, y ∈ X . In particular, the support of v x is independent of x , and hence we write supp v for this common support. (ii) v ( τ ) is real analytic in τ , everywhere except at points τ ∈ supp v where v ( τ ) = 0 . More precisely, there exists C ∼ , such that the derivatives atisfy the bound (cid:107) ∂ kτ v ( τ ) (cid:107) ≤ k ! (cid:16) C (cid:104) v ( τ ) (cid:105) (cid:17) k , ∀ k ∈ N , whenever (cid:104) v ( τ ) (cid:105) > . (iii) The density is uniformly / -Hölder-continuous everywhere, i.e., (cid:107) v ( τ ) − v ( τ ) (cid:107) (cid:46) | τ − τ | / , ∀ τ , τ ∈ R . The comparison relations in these statements depend on the model parameters ρ , L , (cid:107) S (cid:107) L → B , (cid:107) a (cid:107) and Φ . Here we assumed an a priori uniform bound on ||| m ||| R . We remark thatwithout such a bound a regularity result weaker than Theorem 2.4 can still beproven (cf. Corollary 7.4).For simplicity we assume here that ||| m ||| R is bounded. In fact, all the resultsin this paper can be localized on any real interval [ α, β ] , i.e., the statementsapply for τ ∈ [ α, β ] provided ||| m ||| [ α − ε,β + ε ] is bounded for some ε > . Thestraightforward details are left to the reader.The next theorem describes the behavior of the generating density in theregime where the average generating density (cid:104) v (cid:105) is small. We start with definingtwo universal shape functions. Definition 2.5 (Shape functions) . Define Ψ edge : [0 , ∞ ) → [0 , ∞ ) , and Ψ min : R → [0 , ∞ ) , by Ψ edge ( λ ) (2.14a) := (cid:112) (1 + λ ) λ (cid:0) λ + 2 (cid:112) (1 + λ ) λ (cid:1) / + (cid:0) λ − (cid:112) (1 + λ ) λ (cid:1) / + 1 , Ψ min ( λ ) := √ λ ( √ λ + λ ) / + ( √ λ − λ ) / − − . (2.14b)As the names sug-gest, the appropriatelyrescaled versions of theshape functions Ψ edge and Ψ min will describe how v x ( τ + ω ) behaves when τ is an edge of supp v ,i.e., τ ∈ ∂ supp v ,and when τ is a lo-cal minimum of (cid:104) v (cid:105) with (cid:104) v ( τ ) (cid:105) > sufficientlysmall, respectively. Figure 2.1: The two shape functions Ψ edge and Ψ min . 9he next theorem is our main result. Together with Theorem 2.4 it classifiesthe behavior of the generating density of a general bounded solution of the QVE.The theorem generalizes Theorem 2.6 from [AEK16b]. For more details on howthese two results compare, we refer to Section 2.3. Theorem 2.6 (Shape of generating density near its small values) . Assume
A1-3 , and ||| m ||| R ≤ Φ , for some Φ < ∞ . Then the support of the generating measure consists of K (cid:48) ∼ disjoint intervals, i.e., supp v = K (cid:48) (cid:91) i =1 [ α i , β i ] , where β i − α i ∼ , and α i < β i < α i +1 . (2.15) Moreover, for all ε > there exist K (cid:48)(cid:48) = K (cid:48)(cid:48) ( ε ) ∼ points γ , . . . , γ K (cid:48)(cid:48) ∈ supp v such that τ (cid:55)→ (cid:104) v ( τ ) (cid:105) has a local minimum at τ = γ k with (cid:104) v ( γ k ) (cid:105) ≤ ε , ≤ k ≤ K (cid:48)(cid:48) . These minima are well separated from each other and from the edges, i.e., | γ i − γ j | ∼ , ∀ i (cid:54) = j , and | γ i − α j | ∼ , | γ i − β j | ∼ , ∀ i, j . (2.16) Let M denote the set of edges and these internal local minima, M := { α i } ∪ { β j } ∪ { γ k } , (2.17) then small neighborhoods of M cover the entire domain where < (cid:104) v (cid:105) ≤ ε , i.e.,there exists C ∼ such that (cid:8) τ ∈ supp v : (cid:104) v ( τ ) (cid:105) ≤ ε (cid:9) ⊆ (cid:91) i [ α i , α i + C ε ] ∪ (cid:91) j [ β j − C ε , β j ] ∪ (cid:91) k [ γ k − C ε , γ k + C ε ] . (2.18) The generating density is described by expansions around the points of M , i.e.for any τ ∈ M we have v x ( τ + ω ) = v x ( τ ) + h x Ψ( ω ) + O (cid:16) v x ( τ ) + Ψ( ω ) (cid:17) , ω ∈ I , (2.19) where h x ∼ depends on τ . The interval I = I ( τ ) and the function Ψ : I → [0 , ∞ ) depend only on the type of τ according to the following list: • Left edge: If τ = α i , then (2.19) holds with v x ( τ ) = 0 , I = [0 , ∞ ) , and Ψ( ω ) = ( α i − β i − ) / Ψ edge (cid:18) ωα i − β i − (cid:19) , (2.20a) with the convention β − α = 1 . Right edge: If τ = β j , then (2.19) holds with v x ( τ ) = 0 , I = ( −∞ , ,and Ψ( ω ) = ( α j +1 − β j ) / Ψ edge (cid:18) − ωα j +1 − β j (cid:19) , (2.20b) with the convention α K (cid:48) +1 − β K (cid:48) = 1 . • Minimum: If τ = γ k , then (2.19) holds with I = R , and Ψ( ω ) = ρ k Ψ min (cid:18) ωρ k (cid:19) , where ρ k ∼ (cid:104) v ( γ k ) (cid:105) . (2.20c) In case (cid:104) v ( γ k ) (cid:105) = 0 we interpret (2.20c) as its ρ k → limit, i.e., Ψ( ω ) =2 − / | ω | / .All comparison relations depend only on the model parameters ρ , L , (cid:107) S (cid:107) L → B , (cid:107) a (cid:107) and Φ . Figure 2.2 shows an average generating measure which exhibits each of thepossible singularities described by (2.19) and (2.20). Note that the expansions(2.19) become useful for the non-zero minima τ = γ k only when ε > is chosento be so small that the term h x Ψ( ω ) dominates v x ( τ ) which itself is smallerthan ε . Remark . The function ∆ / Ψ edge ( ω/ ∆) describingthe edge shape interpolates between a square root and a cubic root growth withthe switch in the growth rate taking place when its argument becomes of thesize ∆ . Similarly, the function ρ Ψ min ( ω/ρ ) can be seen as a cubic root cusp ω (cid:55)→ | ω | / regularized at scale ρ .Suppose τ is an internal edge with a gap of size ∆ > to the left. As ∆ becomes small, the function λ (cid:55)→ ∆ − / v ( τ + ∆ λ ) approaches the universalshape function Ψ edge up to a λ -independent scaling factor. More precisely,consider a family of data ( a (∆) , S (∆) ) , ∆ ∈ (0 , c ) parameterized by ∆ ∈ (0 , c ) ,such that the supports of the corresponding generating densities v = v (∆) havegaps of size ∆ between opposing internal edges τ = τ (∆)0 and τ − ∆ . If thehypotheses of Theorem 2.6 hold uniformly in ∆ , then lim ∆ ↓ v ( τ + ∆ λ ) v ( τ + ∆ λ ) = Ψ edge ( λ )Ψ edge ( λ ) , ∀ λ , λ > . An analogous statement holds for non-zero local minima and the associateduniversal shape function Ψ min . A simple example of a family of QVEs wherethe gap closes and then becomes a small minima is given in Section 11.6. Remark . We formulated Theorem 2.6 for anarbitrary threshold parameter ε , but it is easy to see that only small values of ε are relevant. In fact, without loss of generality one may assume that ε ∼ isso small that the intervals on the right hand side of (2.18) are disjoint. In this11igure 2.2: Average generating density (cid:104) v (cid:105) when a = 0 and the kernel S xy isa block-constant function as specified with greyscale encoding in the upperright corner. All the possible shapes appear in this example. At the quali-tative level each component v x looks similar. If a is non-zero the v ( τ ) is notnecessarily a symmetric function of τ any more.case the internal minima where (cid:104) v (cid:105) vanishes, i.e., the edges α i , β j and those γ k ’s that correspond to cusps, turn out to be the unique minima within thecorresponding intervals. However, the local minima of (cid:104) v (cid:105) where (cid:104) v (cid:105) (cid:54) = 0 , i.e.,the non-cusp elements of { γ , . . . , γ K (cid:48)(cid:48) } might not be unique even for small ε .In fact, along the proof of Theorem 2.6 we also show (Corollary 9.4) that thesenonzero local minima are either tightly clustered or well separated from eachother in the following sense: If γ, γ (cid:48) ∈ supp v \ ∂ supp v are two local minima of (cid:104) v (cid:105) , then either | γ − γ (cid:48) | (cid:46) min (cid:8) (cid:104) v ( γ ) (cid:105) , (cid:104) v ( γ (cid:48) ) (cid:105) (cid:9) , or | γ − γ (cid:48) | ∼ . In particular, for small ε ∼ , each interval in (2.18) contains at most onesuch cluster of local minima. Within each cluster we may choose an arbitraryrepresentative γ k ; Theorem 2.6 will hold for any such choice.We will now discuss two sufficient and checkable conditions that togetherwith A1-3 imply ||| m ||| R < ∞ , a key input of Theorems 2.4 and 2.6. The firstone involves a regularity assumption on a and the family of row functions, orsimply rows , of S , S x : X → [ 0 , ∞ ) , y (cid:55)→ S xy , x ∈ X , (2.21) 12s elements of L . It expresses that the set of pairs { ( a x , S x ) : x ∈ X } shouldnot have outliers in the sense that, lim ε ↓ inf x ∈ X (cid:90) X π (d y ) ε + ( a x − a y ) + (cid:107) S x − S y (cid:107) = ∞ , (2.22)holds. In other words, this means that no ( a x , S x ) is too different from all theother pairs ( a y , S y ) , y (cid:54) = x . We will see that in case a = 0 , the property (2.22)alone implies a bound for m ( z ) when z is away from zero. When a = 0 thepoint z = 0 is special, and an extra structural condition is needed to ensurethat m (0) is also bounded. In order to state this additional condition we needthe following definitions. Definition 2.9 (Full indecomposability) . A K × K matrix T with non-negativeelements T ij ≥ , is called fully indecomposable (FID) provided that forany subsets I, J ⊂ { , . . . , K } , with | I | + | J | ≥ K , the submatrix ( T ij ) i ∈ I,j ∈ J contains a non-zero entry.The integral operator S : B → B is block fully indecomposable ifthere exist an integer K , a fully indecomposable matrix T = ( T ij ) Ki,j =1 anda measurable partition I := { I j } Kj =1 of X , such that π ( I i ) = 1 K , and S xy ≥ T ij , whenever ( x, y ) ∈ I i × I j , (2.23)for every ≤ i, j ≤ K .The FID property is standard for matrices with non-negative entries [BR97].The most useful properties of FID matrices are listed in Proposition 6.9 andAppendix A.3 below. With these definitions we have the following qualitativeresult on the boundedness of m . Theorem 2.10 (Qualitative uniform bounds) . Suppose that in addition to A1 , A2 and (2.22) , either of the following holds: (i) a = 0 and S is block fully indecomposable; (ii) S satisfies A3 , and inf (cid:26) (cid:104) w, Sw (cid:105)(cid:104) w (cid:105) : w ∈ B , w x ≥ (cid:27) > . (2.24) Then the solution of the QVE is uniformly bounded, ||| m ||| R < ∞ , and in the case (i) S has the property A3 . In particular, the conclusions of both Theorem 2.4and Theorem 2.6 hold. When a = 0 and X is discrete the full indecomposability of S is not only asufficient but also a necessary condition for the boundedness of m in B . Moreprecisely, in Theorem A.4 we will show that in the discrete setup the QVE isstable and has a bounded solution if and only if S is a fully indecomposablematrix. 13e also remark that A3 and the condition (2.24) imply that S is blockfully indecomposable in the discrete setup. In general, neither implies the otherhowever. In Chapter 6 we present quantitative versions of Theorem 2.10: The-orem 6.1 and Theorem 6.4 correspond to the parts (i) and (ii) of Theorem 2.10,respectively.In the prominent example ( X , π (d x )) = ([0 , , d x ) the condition (2.22) issatisfied if the map x (cid:55)→ ( a x , S x ) : X (cid:55)→ R × L is piecewise / -Hölder contin-uous, in the sense that for some finite partition { I k } of [0 , into non-trivialintervals, the bound | a x − a y | + (cid:107) S x − S y (cid:107) ≤ C | x − y | / , ∀ x, y ∈ I k , (2.25)holds for every k . Furthermore, if S has a positive diagonal, such that S xy ≥ ε { | x − y | ≤ δ } , ∀ x, y ∈ [0 , , (2.26)for some ε, δ > , then it is easy to see that S is block fully indecomposable andalso satisfies (2.24), as well as its quantitative version (6.9) (cf. Chapter 6).Next we discuss the special situation in which the generating measure issupported on a single interval. A sufficient condition for this to hold is that thepairs ( a x , S x ) , x ∈ X , can not be split into two well separated subsets in a sensespecified by the inequality (2.27) below. The following result is a quantitativeversion of Theorem 2.8 in [AEK16b]. Theorem 2.11 (Generating density supported on single interval) . Assume S satisfies A1-3 , and ||| m ||| R ≤ Φ . Then there exists a threshold ξ ∗ ∼ such thatunder the assumption sup A ⊂ X inf x ∈ Ay / ∈ A (cid:16) | a x − a y | + (cid:107) S x − S y (cid:107) (cid:17) ≤ ξ ∗ (2.27) the generating density is supported on a single interval, i.e. supp v = [ α, β ] ,with β − α ∼ , and | α | , | β | ≤ Σ . Moreover, for every < δ < ( β − α ) / , wehave v x ( τ ) (cid:38) δ / , τ ∈ [ α + δ, β − δ ] (2.28a) v x ( α + ω ) = h x ω / + O ( ω ) ω ∈ [0 , δ ] , (2.28b) v x ( β − ω ) = h (cid:48) x ω / + O ( ω ) ω ∈ [0 , δ ] , (2.28c) where h, h (cid:48) ∈ B with h x , h (cid:48) x ∼ . Furthermore, v ( τ ) is uniformly / -Höldercontinuous in τ . Here ρ , L , (cid:107) S (cid:107) L → B , (cid:107) a (cid:107) and Φ are considered the modelparameters. S leads to a generating density that issupported on a single interval. Combining the last two theorems weproved that under the conditions ofTheorem 2.10 on ( a, S ) in addition to(2.27) all conclusions of Theorem 2.11hold. For example, if X = [0 , , a = 0 , and S satisfies A1 and A2 ,it is block fully indecomposable, andthe row functions S x are / -Höldercontinuous on the whole set [0 , , thenthe conclusions (2.28) of Theorem 2.11hold true. Figure 2.3 shows an av-erage generating density correspond-ing to an integral operator S with asmooth kernel when a = 0 . Now we discuss the stability properties of the QVE (2.4). These results arethe cornerstone of the proof of the local law for
Wigner-type random matricesproven in [AEK16c], see Chapter 3 for more details. Fix z ∈ H , and suppose g ∈ B satisfies − g = z + a + Sg + d . (2.29)This equation is viewed as a perturbation of the QVE (2.4) by a "small" function d ∈ B . Our final result provides a bound on the difference between g and theunperturbed solution m ( z ) . The difference will be measured both in strong sense(in B -norm) and in weak sense (integrated against a fixed bounded function). Theorem 2.12 (Stability) . Assume S satisfies A1-3 and ||| m ||| R ≤ Φ , for some Φ < ∞ . Then there exists λ ∼ such that if g, d ∈ B satisfy the perturbed QVE (2.29) for some fixed z ∈ H , then the following holds: (i) Rough stability:
Suppose that for some ε ∈ (0 , , (cid:104) v (Re z ) (cid:105) ≥ ε , or dist( z, supp v ) ≥ ε , (2.30) and g is sufficiently close to m ( z ) , (cid:107) g − m ( z ) (cid:107) ≤ λ ε . (2.31) Then their distance is bounded in terms of d as (cid:107) g − m ( z ) (cid:107) (cid:46) ε − (cid:107) d (cid:107) (2.32a) |(cid:104) w, g − m ( z ) (cid:105)| (cid:46) ε − (cid:107) w (cid:107)(cid:107) d (cid:107) + ε − |(cid:104) J ( z ) w, d (cid:105)| , ∀ w ∈ B , (2.32b) for some z -dependent family of linear operators J ( z ) : B → B , thatdepends only on S and a , and satisfies (cid:107) J ( z ) (cid:107) (cid:46) . Refined stability:
There exist z -dependent families t ( k ) ( z ) ∈ B , k =1 , , depending only on S , and satisfying (cid:107) t ( k ) ( z ) (cid:107) (cid:46) , such that thefollowing holds. Defining (cid:36) ( z ) := dist( z, supp v | R ) (2.33a) ρ ( z ) := (cid:104) v (Re z ) (cid:105) (2.33b) δ ( z, d ) := (cid:107) d (cid:107) + |(cid:104) t (1) ( z ) , d (cid:105)| + |(cid:104) t (2) ( z ) , d (cid:105)| , (2.33c) assume g is close to m ( z ) , in the sense that (cid:107) g − m ( z ) (cid:107) ≤ λ (cid:36) ( z ) / + λ ρ ( z ) . (2.34) Then their distance is bounded in terms of the perturbation as (cid:107) g − m ( z ) (cid:107) (cid:46) Υ( z, d ) + (cid:107) d (cid:107) (2.35a) |(cid:104) w, g − m ( z ) (cid:105)| (cid:46) Υ( z, d ) (cid:107) w (cid:107) + |(cid:104) T ( z ) w, d (cid:105)| , ∀ w ∈ B , (2.35b) for some z -dependent family of linear operators T ( z ) : B → B , thatdepends only on S and a , and satisfies (cid:107) T ( z ) (cid:107) (cid:46) . Here the key controlparameter is Υ( z, d ) := min (cid:26) δ ( z, d ) ρ ( z ) , δ ( z, d ) (cid:36) ( z ) / , δ ( z, d ) / (cid:27) . (2.36) The comparison relations depend on ρ , L , (cid:107) S (cid:107) L → B , (cid:107) a (cid:107) and Φ . Note that the existence of g solving (2.29) for a given d is part of the assump-tions of Theorem 2.12. In Proposition 7.5 we will actually prove the existenceand uniqueness of g close to m provided d is sufficiently small. An importantaspect of the estimates (2.32) and (2.35) is that the upper bounds depend onlyon the unperturbed problem, i.e., on z , a and S , possibly through m ( z ) , apartfrom the explicit dependence of d . They do not depend on g .The condition (2.34) of (ii) in the preceding theorem becomes increasinglyrestrictive when z approaches points in supp v where v takes small values. Astronger but less transparent perturbation estimate is given as Proposition 10.1below.The guiding principle behind these estimates is that the norm bounds (2.32a)and (2.35a) are linear in (cid:107) d (cid:107) , while the bounds (2.32b) and (2.35b) for theaverage of g − m are quadratic in (cid:107) d (cid:107) and linear in a specific average in d .The motivation behind the average bounds is that in the random matrix theory(cf. Chapter 3) the perturbation d will be random. In fact, d will be subjectto the fluctuation averaging mechanism, i.e., its (weighted) average is typicallycomparable to (cid:107) d (cid:107) in size. In the part (ii) of the theorem we see how thestability estimates deteriorate as z approaches the part of the real line where (cid:104) v (cid:105) becomes small, in particular near the edges of supp v .Another trivial application of our general stability result is to show that theQVE (2.4) is stable under perturbations of a and S .16 emark a and S ) . Suppose S and T are two integraloperators satisfying A1-3 and a, b ∈ B are real valued. Let m and g be theunique solutions of the two QVE’s − m = z + a + Sm and − g = z + b + T g .
Then g can be considered as a solution of the perturbed QVE (2.29), with d := ( b − a ) + ( T − S ) g . Thus if ||| m ||| R < ∞ , then Theorem 2.12 may be used to control g − m in termsof b − a and T − S . 17 .3 Relationship between Theorem 2.6 and The-orem 2.6 of [AEK16b] Theorem 2.6 is a quantitative generalization of The-orem 2.6 of [AEK16b]. We comment on the differ-ences between the two results. The main novelty inTheorem 2.6 is that it provides a precise descrip-tion of the generating density around the expansionpoints τ in an environment whose size is compa-rable to . Moreover, its statement is uniform inthe operator S and the function a , given the modelparameters. In [AEK16b], on the other hand, theoperator S is fixed and only asymptotically small ex-pansion environments are considered. Theorem 2.6also provides explicit quantitative error bounds interms of the model parameters.To illustrate the distinction between the tworesults we consider a fixed a , and a continuous one-parameter family of operators S = S ( δ ) with thefollowing properties:1. The family S ( δ ) satisfies A1-3 uniformly in δ .2. The corresponding solutions m ( δ ) are uni-formly bounded, sup δ ||| m ( δ ) ||| R ≤ Φ .3. There is an expansion point τ ( δ ) , dependingcontinuously on δ , such that (cf. Figure 2.4)(a) At a critical value δ = δ c the generat-ing density corresponding to S ( δ c ) has acubic root cusp at τ = τ ( δ c ) , i.e., theexpansion point τ is a minimum in thesense of (2.20c) and (cid:104) v ( τ ) (cid:105) = 0 .(b) For δ > δ c the expansion point τ = τ ( δ ) is a minimum in the sense of (2.20c) ofthe generating density corresponding to S ( δ ) with (cid:104) v ( τ ) (cid:105) > .(c) For δ < δ c the expansion point τ = τ ( δ ) is a left edge in the sense of (2.20a) of thegenerating density corresponding to S ( δ ) . Figure 2.4: Differ-ent singularity shapesemerge at τ by varying δ .We refer to Section 11.6 for an explicit example of such a family of operators S = S ( δ ) . The results of [AEK16b] analyze the situation only for a fixed valueof the parameter δ and they are restricted to a description of the generatingdensity in asymptotically small expansion environments. In other words, in18ach case (2.20a), (2.20b) and (2.20c) only the limiting behavior as ω → ofthe function Ψ is tracked. Indeed, Theorem 2.6 reduces to Theorem 2.6 in[AEK16b] containing the following statements:(a) At the critical value δ = δ c we have v x ( τ + ω ) = 2 − / h x | ω | / (1 + o (1)) as ω → .(b) For any fixed δ > δ c we have v x ( τ + ω ) = v x ( τ )(1 + o (1)) as ω → .(c) For any fixed δ < δ c we have v x ( τ + ω ) = / h x ω / (1 + o (1)) as ω ↓ .Here, ∆ > is the length of the gap in the support of the generatingdensity whose right boundary is τ .In particular, the statement (b) does not contain any interesting information andthus expansion points τ of the minimum type with (cid:104) v ( τ ) (cid:105) > were even notconsidered in Theorem 2.6 of [AEK16b]. In Theorem 2.6 of the current paper,however, the description is uniform in δ and covers an expansion neighborhoodaround τ whose size is comparable to , i.e., it describes the shape of Ψ( ω ) for all | ω | ≤ c for some constant c ∼ . Thus, the new result resolves the twouniversal shape functions from (2.14) and reveals how these functions give riseto a continuous one-parameter family of shapes interpolating between them. Inparticular, it shows how the cusp singularity emerges when a gap closes or whenthe value of v at a local minimum drops down to zero. Indeed, as the length ofthe gap ∆ in the support of the generating density at τ shrinks (as δ ↑ δ c for theexample family S ( δ ) ) the shape function Ψ( ω ) = ∆ / Ψ edge (∆ − ω ) approachesthe cusp shape Ψ( ω ) = 2 − / | ω | / . On the other hand, as the value ρ ∼ (cid:104) v ( τ ) (cid:105) in the shape function Ψ( ω ) = ρ Ψ min ( ρ − ω ) at a local minimum approaches zero( δ ↓ δ c for the family S ( δ ) ), the cubic root cusp emerges as well. The followingtable summarizes the differences between the current Theorem 2.6 and Theorem2.6 of [AEK16b]. Theorem 2.6 in [AEK16b] Current Theorem 2.6Input parameters: a, S fixed model parametersExpansion points τ : { α i } ∪ { β j } { α i } ∪ { β j } ∪ { γ k } Expansion environment: | ω | (cid:28) | ω | (cid:46) In this section we will explain and motivate the basic steps leading to our mainresults.
Stieltjes transform representation, L - and uniform bounds: It is astructural property of the QVE that its solution admits a representation as theStieltjes transform of some generating measure on the real line (cf. (2.8)). Thisrepresentation implies that m can be fully reconstructed from its own imaginarypart near the real line. 19rom the Stieltjes transform representation of m a trivial bound, | m x ( z ) | ≤ (Im z ) − , directly follows. A detailed analysis of the QVE near the real axis,however, requires bounds that are independent of Im z as its starting point.When a = 0 and z is bounded away from zero the L -bound (2.9) meets thiscriterion. The estimate (2.9) is a structural property of the QVE as well in thesense that it follows from positivity and symmetry of S alone, and thereforequantitative assumptions such as A2 and A3 are not needed. This L -boundis derived from spectral information about a specific operator F = F ( z ) , con-structed from the solution m = m ( z ) , that appears naturally when taking theimaginary part on both sides of the QVE. Indeed, (2.4) implies Im m | m | = | m | Im z + F Im m | m | , F u := | m | S ( | m | u ) . (2.37)As Im z approaches zero we may view this as an eigenvalue equation for thepositive symmetric linear operator F . In the limit this eigenvalue equals and f = Im m/ | m | is the corresponding eigenfunction, provided Im m does notvanish. The Perron-Frobenius theorem, or more precisely, its generalizationto compact operators, the Krein-Rutman theorem, implies that this eigenvaluecoincides with the spectral radius of F . This, in turn, implies the L -bound on m , when a = 0 . These steps are carried out in detail at the end of Chapter 4.In fact, the norm of F ( z ) , as an operator on L , approaches if and only if z approaches the support of the generating measure. Otherwise it stays below . When a (cid:54) = 0 this spectral bound still holds for F , however, it does notautomatically yield useful L -estimates on m ( z ) when | z | ≤ (cid:107) a (cid:107) . In order, toobtain an L -bound in this case as well, we need to assume more about S . InChapter 6 it is shown that the condition (2.24), or its quantitative version B2 on p. 55, together with the spectral bound on F , yield an L -bound on m .Requiring the additional regularity condition (2.22) on S enables us to im-prove the L -bound on m to a uniform bound (Proposition 6.6). When a = 0 the point z = 0 requires a special treatment, because the structural bound (cid:107) m ( z ) (cid:107) ≤ / | z | becomes ineffective. The block fully indecomposability condi-tion is an essentially optimal condition (Theorem A.4) to ensure the uniformboundedness of m in a vicinity of z = 0 when a = 0 . The uniform bounds area prerequisite for most of our results concerning regularity and stability of thesolution of the QVE. We consider finding quantitative uniform bounds on m asan independent problem, that is addressed in Chapter 6. Stability in the region where Im m is large: Stability properties of theQVE under small perturbations are essential, not just for applications in randommatrix theory (cf. Chapter 3), but also as tools to analyze the regularity of thesolution m ( z ) ∈ B + as a function of z . Indeed, the stability of the QVE trans-lates directly to regularity properties of the generating measure as described byTheorem 2.4. The stability of the solution deteriorates as Im m becomes small.This happens around the expansion points in M from Theorem 2.6In order to see this deterioration of the stability, let us suppose that fora small perturbation d ∈ B , the perturbed QVE has a solution g ( d ) which20epends smoothly on d , − g ( d ) = z + a + Sg ( d ) + d . (2.38)Indeed, the existence and uniqueness of such a function d (cid:55)→ g ( d ) is shown inProposition 7.5 as long as both d and g − m are sufficiently small. For d = 0 weget back our original solution g (0) = m , with m = m ( z ) . We take the functionalderivative with respect to d on both sides of the equation. In this way we derivea formula for the (Fréchet-)derivative Dg (0) , evaluated on some w ∈ B : (1 − m S ) Dg (0) w = m w . (2.39)This equation shows that the invertibility of the linear operator − m S isrelevant to the stability of the QVE. Assuming uniform lower and upper boundson | m | , the invertibility of − m S is equivalent to the invertibility of thefollowing related operator: B := U − F = | m | m (1 − m S ) | m | , U w := | m | m w . Here, | m | on the right of S is interpreted as a multiplication operator by | m | .Similarly, U is a unitary multiplication operator and F was introduced in (2.37).Away from the support of the generating measure the spectral radius of F staysbelow and the invertibility of B is immediate. On the support of the generatingmeasure the spectral radius of F equals . Here, the fundamental bound on theinverse of B is (cid:107) B − (cid:107) (cid:46) (cid:104) Im m (cid:105) − , (2.40)apart from some special situations (cf. Lemma 5.9).Let us understand the mechanism that leads to this bound in the simplestcase, namely when x (cid:55)→ m x ( z ) is a constant function, e.g., when a = 0 and (cid:104) S x (cid:105) = 1 , so that each component equals m sc ( z ) from (1.4). In this situation,the operator U is simply multiplication by a complex phase, U = e i ϕ with ϕ ∈ ( − π, π ] . The uniform bounds on m ensure that the operator F inheritscertain properties from S . Among these are the conditions A2 and A3 . Fromthese two properties we infer a spectral gap ε > , Spec( F ) ⊆ [ − ε, − ε ] ∪ { } , on the support of the generating measure. We readily verify the following boundon the norm of the inverse of B : (cid:107) B − (cid:107) L → L ≤ (cid:40) | e i ϕ − | − ∼ (cid:104) Im m (cid:105) − if ϕ ∈ [ − ϕ ∗ , ϕ ∗ ] ; ε − otherwise.Here, ϕ ∗ ∈ [0 , π/ is the threshold defined through cos ϕ ∗ = 1 − ε/ , where thespectral radius (cid:107) F (cid:107) L → L becomes more relevant for the bound than the spectral21ap (cf. Lemma 5.7). Similar bounds for the special case, when m x = m sc isconstant in x and equals the Stieltjes transform m sc of the semicircle law inevery component first appeared in [EYY11a].The bound (2.40) on the inverse of B implies a bound on the derivative Dg (0) from (2.39). For a general perturbation d this means that the QVE isstable wherever the average generating measure is not too small. If d is chosento be a constant function d x = z (cid:48) − z then this argument yields the bound forthe difference m ( z (cid:48) ) − m ( z ) , as g ( z (cid:48) − z ) = m ( z (cid:48) ) . This can be used to estimatethe derivative of m ( z ) with respect to z and to prove existence and Hölder-regularity of the Lebesgue-density of the generating measure. In particular, theregularity is uniform in Im z and hence we can extend the solution of the QVEto the real axis. This analysis is carried out in Chapters 5 and 7. Stability in the regime where Im m is small: The bound (2.40) becomesineffective when (cid:104) Im m (cid:105) approaches zero. In fact, the norm of B − divergesowing to a single isolated eigenvalue, β ∈ C , close to zero. This point is associ-ated to the spectral radius of F , and the corresponding eigenvector, B b = β b ,is close to the Perron-Frobenius eigenvector of F , i.e., b = f + O ( (cid:104) Im m (cid:105) ) , with F f = f . The special direction b , in which B − becomes unbounded, is treatedseparately in Chapter 9. It is split off from the derivative Dg (0) in the stabilityanalysis. The coefficient of the component g ( d ) − m in the bad direction b , isgiven by the formula Θ( d ) := (cid:10) b, g ( d ) − m (cid:11) (cid:104) b (cid:105) . Chapter 8 is concerned with deriving a cubic equation for Θ( d ) and expandingits coefficients in terms of (cid:104) Im m (cid:105) (cid:28) at the edge. Universal shape of v near its small values: In this regime understandingthe dependence of the solution g ( d ) of (2.38), is essentially reduced to under-standing the scalar quantity Θ( d ) . This quantity satisfies a cubic equation (cf.Proposition 8.2), in which the coefficients of the non-constant terms dependonly on the unperturbed solution m . In particular, we can follow the depen-dence of m x ( z ) on z ∈ R by analyzing the solution of this equation by choosing z := τ ∈ R and d x := τ − τ , a real constant function. The special structureof the coefficients of the cubic equation, in combination with specific selectionprinciples, based on the properties of the solution of the QVE, allows only for afew possible shapes that the solution τ (cid:55)→ Θ( τ − τ ) of the cubic equation mayhave. This is reflected in the universal shapes that describe the growth behaviorof the generating density at the boundary of its support. In Chapter 9 we willanalyze the three branches of solutions for the cubic equation in detail and selectthe one that coincides with Θ . This will complete the proof of Theorem 2.6. Optimal Stability around small minima of (cid:104) v (cid:105) : For the random matrixtheory we need optimal stability properties of the perturbation g ( d ) around g (0) = m for a random perturbation (cf. Chapter 3). This is achieved inChapter 10 by describing the coefficients of the cubic more explicitly based on22he shape analysis. All the necessary results are collected in Proposition 10.1.These technical results generalize Theorem 2.12.23 hapter 3 Local laws for large randommatrices
The QVE plays a fundamental role in the theory of large random matrices.First, it provides the only known effective way to determine the asymptoticeigenvalue density for prominent matrix ensembles as described in the introduc-tion (cf. Section 3 of [AEK16b] for details). Second, the QVE theory is essentialwhen establishing local laws for the distribution of the eigenvalues at the scalecomparable to the individual eigenvalue spacings for so-called
Wigner-type ma-trices. Here we explain how our results can be utilized for this purpose. Since alltechnical details are already carried out in [AEK16c] we highlight the structureof the proofs in the simplest possible setup by showing how the probabilistic es-timates and the stability properties of the QVE can be turned into very preciseprobabilistic bounds on the resolvent elements of the random matrix.Let us recall from [AEK16c] the following definition.
Definition 3.1 (Wigner-type random matrix) . A real symmetric or complexhermitian N × N random matrix H = ( h ij ) Ni,j =1 is called Wigner-type , if ithas(i) Centred entries: E h ij = 0 ;(ii) Independent entries: ( h ij : 1 ≤ i ≤ j ≤ N ) are independent;(iii) Mean-field property: The variance matrix S = ( s ij ) Ni,j =1 , s ij := E | h ij | ,satisfies ( S L ) ij ≥ ρN and s ij ≤ S ∗ N , ≤ i, j ≤ N , (3.1)for some parameters ρ, L, S ∗ < ∞ .If in addition to (i)-(iii) the variance matrix is doubly stochastic, i.e., (cid:80) j s ij =1 for each i , and (3.1) holds with L = 1 , then H is called a generalized Wignermatrix (first introduced in [EYY11a]).24 given variance matrix S defines a QVE through X := { , , . . . , N } , π ( A ) := | A | N , a = 0 , ( Sw ) i := N (cid:88) j =1 s ij w j , (3.2)where the subset A ⊂ { , . . . , N } and the function w : X → C are arbitrary. Thekernel of the operator S : B → B is related to the variances by S ij := N s ij . Inparticular, if (3.1) is assumed, then the operator S satisfies A1 , as well as A2 and A3 with parameters ρ, L and (cid:107) S (cid:107) L → B ≤ S ∗ , respectively.A local law for H roughly states that the density of the eigenvalues λ ≤ . . . ≤ λ N of H is predicted by the associated QVE through ρ ( z ) := 1 π (cid:104) Im m ( z ) (cid:105) , (3.3)all the way down to the optimal scale Im z (cid:29) N − , just above the typicaleigenvalue spacing. Moreover, the local law implies that the eigenvectors arecompletely delocalized , i.e., no component of an (cid:96) -normalized eigenvector of H is much larger than N − / with very high probability (cf. Corollary 1.14 of[AEK16c]). A local law is most generally stated in term of the entries of theresolvent G ( z ) := ( H − z ) − , z ∈ H . (3.4)The following is a simplified version of the main local law theorem of [AEK16c].It states that G ( z ) approaches the diagonal matrix determined by the solution m ( z ) of the QVE, provided the imaginary part of the spectral parameter z isslightly larger than the eigenvalue spacing, N − , inside the bulk of the spectrum.Indeed, denoting D ( N ) γ := (cid:8) z ∈ C : N γ − < Im z ≤ Σ (cid:9) . (3.5)where γ > and Σ > is from (2.7), the theorem reads: Theorem 3.2 (Entrywise local law from [AEK16c]) . Let H be a Wigner-typerandom matrix, and suppose the associated QVE (3.2) has a bounded solution m , with ||| m ||| R ≤ Φ . If additionally, the moments of H are bounded by thevariances, E | h ij | p ≤ µ p s pij , ∀ p ≥ , (3.6) then the entries G ij ( z ) of the resolvent (3.4) satisfy for every φ, γ, p > , P (cid:40) ∃ z ∈ D ( N ) γ s.t. (cid:12)(cid:12) G ij ( z ) − δ ij m i ( z ) (cid:12)(cid:12) > N φ √ N Im z (cid:41) ≤ C ( φ, γ, p ; ξ S , µ ) N p , where the function C ( · , · , · ; ξ S , µ ) < ∞ depends on H only through the param-eters ξ S := ( ρ, L, S ∗ , Φ) and µ = ( µ p : p ≥ .
25e stress that the error bound in the local law does not depend on thevariance matrix through anything else than the parameters ρ , L , S ∗ , and Φ . Ifthe operator S also satisfies the quantitative versions of the assumptions (i) ofTheorem 2.10, then the implicit constant Φ can also be effectively bounded interms of the variance matrix using a few additional model parameters appearingin the hypotheses of Theorem 6.1 below.It is also shown in [AEK16c] that under the conditions of the previous theo-rem an averaged local law holds with an improved error bound. More precisely,for any non-random weights w k , and φ > , we have N (cid:12)(cid:12)(cid:12)(cid:12)(cid:88) k w k ( G kk ( z ) − m k ( z )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ N φ max i | w i | E N ( z ) N Im z , (3.7)with very high probability for sufficiently large N . Here the error term E N ( z ) is O (1) , except when z approaches an asymptotically small non-zero minimum oran asymptotically small gap in supp v (cf. formulas (1.21) and (1.23)-(1.25) in[AEK16c] for details). In particular, choosing w k = 1 in (3.7) and consideringthe spectral parameters z in the bulk of the spectrum, so that (cid:104) v (Re z ) (cid:105) > ,we find for every φ > , (cid:12)(cid:12)(cid:12) N Tr G ( z ) − (cid:104) m ( z ) (cid:105) (cid:12)(cid:12)(cid:12) ≤ N φ N Im z , (3.8)with very high probability. This estimate is the starting point for proving the local bulk universality for eigenvalues of H . For more details see Theorem 1.7of [AEK16c].All these result have been originally obtained for generalized Wigner matricesin a sequence of papers [EYY11a, EYY11b, EYY12], see [EKYY13b] for a sum-mary. The main difference is that for generalized Wigner matrices the limitingdensity is given by the explicit Wigner semicircle law (1.4), while Wigner-typematrices have a quite general density profile that is known only implicitly fromthe solution of the QVE using (3.3). In particular, the density may have cubicroot singularities (cf. Theorem 2.6), as opposed to two square root singularitiesof the semicircle law, and these new kind of singularities require a new proof forthe local law. In order to see why Theorem 3.2 should hold we first apply the Schur comple-ment formula for the diagonal entries of the resolvent (3.4) to get − G kk ( z ) = z − h kk + ( k ) (cid:88) i,j h ki G ( k ) ij ( z ) h jk , (3.9)where (cid:80) ( k ) i,j denotes the sum over all indices i, j not equal to k , and G ( k ) ij ( z ) arethe entries of the resolvent of the matrix obtained by setting the k -th row and26he k -th column of H equal to zero. Replacing the terms on the right hand sideof (3.9) by their partial averages w.r.t. the k -th row and column, and regardingthe rest as perturbations, we arrive at a perturbed QVE, − G kk ( z ) = z − E h kk + 1 N N (cid:88) i =1 S ki G ii ( z ) + d k ( z ) , (3.10)for the diagonal entries of the resolvent G kk ( z ) . Here the random error is givenby d k ( z ) = ( k ) (cid:88) i (cid:54) = j h ki G ( k ) ij ( z ) h jk + ( k ) (cid:88) i (cid:0) | h ki | − E | h ki | (cid:1) G ( k ) ii ( z )+ 1 N ( k ) (cid:88) i S ki ( G ( k ) ii ( z ) − G ii ( z )) − ( h kk − E h kk ) − S kk N G kk ( z ) . (3.11)Setting a k = − E h kk we identify (3.10) with the perturbed QVE (2.29). Forthe sake of simplicity, we consider only the case E h kk = 0 here.Since G ( k ) ( z ) , by definition does not depend on the k -th row/column of H ,the centered terms h ki , h jk and ( | h ki | − E | h ki | ) are independent of G ( k ) ( z ) in(3.11). Therefore the first term on the right hand side of (3.11) can be controlledby the standard large deviation estimate (cf. Appendix B of [EKYY13a]) of theform P (cid:40) (cid:12)(cid:12)(cid:12) (cid:88) i (cid:54) = j a ij X i X j (cid:12)(cid:12)(cid:12) ≥ N κ (cid:88) i (cid:54) = j | a ij | (cid:41) ≤ C ( κ, q ) N q . (3.12)Here X i ’s are independent and centered random variables with finite moments,and the exponents κ, q > are arbitrary. A similar bound holds for the secondterm on the right hand side of (3.11).Lemma 2.1 in [AEK16c], states that if Λ( z ) := N max i,j =1 (cid:12)(cid:12) G ij ( z ) − δ ij m i ( z ) (cid:12)(cid:12) (3.13)satisfies a rough a priori estimate, then the perturbation d k ( z ) can be shownto be very small using standard large deviation estimates, such as (3.12), andstandard resolvent identities. A simplified version of this lemma is formulatedas follows: Lemma 3.3 (Probabilistic part for simplified local law) . Under the assumptionsof Theorem 3.2, there exist a threshold λ > and constants C ( κ, q ) < ∞ , forany κ, q > , such that for any z ∈ H P (cid:40) (cid:18) (cid:107) d ( z ) (cid:107) + N max i,j =1 i (cid:54) = j | G ij ( z ) | (cid:19) (cid:8) Λ( z ) ≤ λ (cid:9) > N κ δ N ( z ) (cid:41) ≤ C ( κ, p ) N q , (3.14) 27 here (cid:107) · (cid:107) denotes the supremum norm, and δ N ( z ) := 1 √ N Im z + 1 √ N . (3.15)
Here λ and the constants C ( κ, p ) are independent of z . They depend on therandom matrix H only through the parameters ξ S , µ defined in Theorem 3.2. We we will now show how to prove the entrywise local law, Theorem 3.2,in the special case where the spectral parameter z satisfies the bulk assumption (2.30) for some ε > . The proof demonstrates the general philosophy of howthe non-random stability results for the QVE, such as Theorem 2.12, are usedtogether with probabilistic estimates, such as Lemma 3.3 above. Our estimateswill deteriorate as the lower bound ε in the bulk assumption approaches zero. Inorder to get the local law uniformly in ε a different and much more complicatedargument (cf. Section 4 of [AEK16c]) is needed. In particular, Theorem 2.12must be replaced by its more involved version, Proposition 10.1.In order to obtain the averaged local law (3.7), under the bulk assumption,the componentwise estimate (2.32a) must be replaced by the averaged estimate(2.32b), which bounds G kk ( z ) − m k ( z ) , in terms of a weighted average of d k ( z ) .The improvement comes from the fluctuation averaging mechanism introducedin [EKYY13b, EYY11b]. In fact, Theorem 3.5 of [AEK16c] shows that (cid:104) w, d ( z ) (cid:105) ,for any non-random w ∈ B , is typically of size (cid:107) w (cid:107) (cid:107) d ( z ) (cid:107) , and hence muchsmaller than the trivial bound (cid:107) w (cid:107)(cid:107) d (cid:107) used in the entrywise local law. For theaveraged bounds, the bulk assumption (2.30) can be removed as well by usingProposition 10.1 in place of Theorem 2.12. Proof of Theorem 3.2 in the bulk.
Let us fix τ ∈ R such that (2.30)holds for all z on the line L := τ + i [ N γ − , N ] . (3.16)We will also fix an arbitrary γ > . Clearly, it suffices to prove the local lawonly when N is larger than some threshold N = N ( φ, γ, p ) < ∞ , dependingonly on ξ S , µ , in addition to the arbitrary exponents φ, γ, p > .Combining (3.14) with the stability of the QVE under the perturbation d ( z ) ,Theorem 2.12, we obtain (cid:18) N max k =1 | G kk ( z ) − m k ( z ) | (cid:19) (cid:8) Λ( z ) ≤ λε (cid:9) ≤ C ε (cid:107) d ( z ) (cid:107) . (3.17)Here the indicator function guarantees that the part (i) of Theorem 2.12 isapplicable. The constant λ ∼ is taken from that theorem, while C ∼ is thehidden constant in (2.32a).Combining (3.17) with (3.14) we see that for every κ, q > , and every fixed z ∈ H , there exists an event Ω κ,q ( z ) , of very high probability P ( Ω κ,q ( z )) ≥ − C ( κ, q ) N − q , (3.18) 28uch that for a sufficiently large threshold N and every N ≥ N we get Λ( z ; ω ) { Λ( z ; ω ) ≤ λ ∗ } ≤ N κ δ N ( z ) , ∀ ω ∈ Ω κ,q ( z ) , (3.19)where λ ∗ := min { λ , λε } .The event Ω κ,q ( z ) depends on the spectral point z ∈ L . As a next step wereplace the uncountable family of events Ω κ,q ( z ) , z ∈ L , in (3.19) by a singleevent, that covers all z ∈ L . To this end, we use the regularity of the resolventelements and of the solution to the QVE in the spectral variable z . Indeed, theyare both Stieltjes transforms of probability measures (cf. (2.8)), and thus theirderivatives are uniformly bounded by (Im z ) − ≤ N when z ∈ L . In particular,it follows that | Λ( z (cid:48) ) − Λ( z ) | ≤ N | z (cid:48) − z | , z, z (cid:48) ∈ L . (3.20)Let L N consist of N evenly spaced points on L , such that the N − -neighborhoodof L N covers L . Combining (3.20) and (3.19) we see that for any φ, p > , theintersection event, Ω φ,p := (cid:92) z ∈ L N Ω φ/ ,p +5 ( z ) , (3.21)has the properties P ( Ω φ,p ) ≥ − C ( φ, p ) N − p (3.22a) Λ( z ; ω ) { Λ( z ; ω ) ≤ λ ∗ } ≤ N φ δ N ( z ) , ∀ ( z, ω ) ∈ L × Ω φ,p . (3.22b)Here C ( φ, p ) := C ( φ/ , p + 5) , with C ( · , · ) taken from (3.18). In order toprove (3.22b) pick an arbitrary pair ( z, ω ) ∈ L × Ω φ,p , and set κ := φ/ and q := p + 5 . If z ∈ L N , then the claim follows directly from (3.19) and (3.21). Inthe case z / ∈ L N , let z (cid:48) ∈ L N be such that | z (cid:48) − z | ≤ N − . Suppose now that Λ( z ; ω ) ≤ λ ∗ . By the continuity (3.20) we see that Λ( z (cid:48) ; ω ) ≤ λ ∗ , and thus(3.19) yields Λ( z (cid:48) ; ω ) ≤ N κ δ N ( z (cid:48) ) . Using (3.20) together with δ N ( z ) ≥ N − / and | δ N ( z ) − δ N ( z (cid:48) ) | ≤ N / | z − z (cid:48) | we get Λ( z ; ω ) ≤ N κ δ N ( z ) . This proves(3.22b).The proof of the local law is now completed by showing that the indicatorfunction is identically equal to one in (3.22b) for ( z, ω ) ∈ L × Ω φ,p , provided φ < γ/ . Indeed, if N is so large that N φ − γ/ < λ ∗ / , then N φ δ N ( z ) < λ ∗ / ,for N ≥ N , and thus the bound (3.22b) implies Λ( z ; ω ) / ∈ (cid:20) λ ∗ , λ ∗ (cid:21) , ∀ ( z, ω ) ∈ L × Ω φ,p . Fix ω ∈ Ω φ,p . Since z (cid:55)→ Λ( z ; ω ) is continuous, the set Λ( L ; ω ) is simplyconnected. Therefore it is contained either in [0 , λ ∗ / , or in [ λ ∗ , ∞ ) . The latterpossibility is excluded by considering the point z := τ + i N ∈ L . Indeed, from(3.4) and the Stieltjes transform representations it follows that | G ij ( z ; ω ) − δ ij m i ( z ) | ≤ z = 2 N , i, j = 1 , . . . , N . N is so large that /N < λ ∗ / , we see Λ( z ; ω ) < λ ∗ / . Thiscompletes the proof of Theorem 3.2 for spectral parameters z satisfying the bulkcondition (2.30). 30 hapter 4 Existence, uniqueness and L -bound This chapter contains the proof of Theorem 2.1. Namely assuming, • S satisfies A1 ,we show that the QVE (2.4) has a unique solution, whose components m x areStieltjes transforms (cf. (2.8)) of x − dependent probability measures, supportedon the interval [ − Σ , Σ ] . We also show that if a = 0 , then m ( z ) ∈ L , whenever z (cid:54) = 0 (cf. (2.9)). The existence and uniqueness part of Theorem 2.1 is provenby considering the QVE as a fixed point problem in the space B + . The choiceof an appropriate metric on B + is suggested by the general theory of Earleand Hamilton [EH70]. A similar line of reasoning for the proof of existenceand uniqueness results that are close to the one presented here has appearedbefore (see e.g. [AZ05, HFS07, KLW13, FHS07]). The structural L -estimate inSection 4.2 is the main novelty of this chapter.For the purpose of defining the correct metric on B + we use the standardhyperbolic metric d H on the complex upper half plane H . This metric has theadditional benefit of being invariant under z (cid:55)→ − z − , which enables us toexchange the numerator and denominator on the left hand side of the QVE.We start by summarizing a few basic properties of d H . These will be ex-pressed through the function D ( ζ, ω ) := | ζ − ω | (Im ζ )(Im ω ) , ∀ ζ, ω ∈ H , (4.1)which is related to the hyperbolic metric through the formula D ( ζ, ω ) = 2 (cosh d H ( ζ, ω ) − . (4.2) Lemma 4.1 (Properties of hyperbolic metric) . The following three propertieshold for D : . Isometries: If ψ : H → H , is a linear fractional transformation, of theform ψ ( ζ ) = αζ + βγ ζ + µ , (cid:20) α βγ µ (cid:21) ∈ SL ( R ) , then D (cid:0) ψ ( ζ ) , ψ ( ω ) (cid:1) = D ( ζ, ω ) .
2. Contraction: If ζ , ω ∈ H are shifted in the positive imaginary direction by λ > then D ( ζ + i λ, ω + i λ ) = (cid:16) λ Im ζ (cid:17) − (cid:16) λ Im ω (cid:17) − D ( ζ, ω ) . (4.3)
3. Convexity: Suppose (cid:54) = φ ∈ B ∗ is a bounded non-negative linear func-tional on B , i.e., φ ( u ) ≥ for all u ∈ B with u ≥ . Let u, w ∈ B + with imaginary parts bounded away from zero, inf x Im u x , inf x Im w x > .Then D (cid:0) φ ( u ) , φ ( w ) (cid:1) ≤ sup x ∈ X D ( u x , w x ) . (4.4) Proof.
Properties 1 and 2 follow immediately from (4.2) and (4.1). It remainsto prove Property 3. The functional φ is non-negative. Thus, | φ ( w ) | ≤ φ ( | w | ) for all w ∈ B . Therefore, D (cid:0) φ ( u ) , φ ( w ) (cid:1) ≤ φ ( | u − w | ) φ (Im u ) φ (Im w ) = φ (cid:0) (Im u ) / (Im w ) / D ( u, w ) / (cid:1) φ (Im u ) φ (Im w ) , (4.5)where we used the definition of D from (4.1) two times. We apply a version ofJensen’s inequality for bounded linear, non-negative and normalized functionalson B to estimate further, φ (cid:0) (Im u ) / (Im w ) / D ( u, w ) / (cid:1) φ (cid:0) (Im u ) / (Im w ) / (cid:1) ≤ φ (cid:0) (Im u ) / (Im w ) / D ( u, w ) (cid:1) . (4.6)We combine (4.5) with (4.6) and use the non-negativity of φ to estimate D ( u, w ) ≤ sup x ∈ X D ( u x , w x ) inside its argument, D (cid:0) φ ( u ) , φ ( w ) (cid:1) ≤ φ (cid:0) (Im u ) / (Im w ) / (cid:1) φ (Im u ) φ (Im w ) sup x ∈ X D ( u x , w x ) . (4.7)Finally we use φ ( g / h / ) ≤ φ ( g ) + φ ( h ) for the choice g := Im u/φ (Im u ) and h := Im w/φ (Im w ) to show that the fraction on the right hand side of (4.7) isnot larger than . This finishes the proof of (4.4).32n order to show existence and uniqueness of the solution of the QVE forgiven S and a , we see that for any fixed z ∈ H , a solution m = m ( z ) ∈ B + of(2.6) is a fixed point of the map Φ( · ; z ) : B + → B + , Φ( u ; z ) := − z + a + Su . (4.8)Let us fix a constant η ∈ (0 , min { , / (cid:107) a (cid:107)} ) such that z lies in the domain H η := (cid:8) z ∈ H : | z | < η − , Im z > η (cid:9) . (4.9)We will now see that Φ( · ; z ) is a contraction on the subset B η := (cid:26) u ∈ B + : (cid:107) u (cid:107) ≤ η , inf x ∈ X Im u x ≥ η (2 + (cid:107) S (cid:107) ) (cid:27) , (4.10)equipped with the metric d ( u, w ) := sup x ∈ X d H ( u x , w x ) , u, w ∈ B η . (4.11)On B η the metric d is equivalent to the metric induced by the uniform norm(2.2) of B . Since B η is closed in the uniform norm metric it is a completemetric space with respect to d . Lemma 4.2 ( Φ is contraction) . For any z ∈ H η , the function Φ( · ; z ) maps B η into itself and satisfies sup x ∈ X D (cid:16)(cid:0) Φ( u ; z ) (cid:1) x , (cid:0) Φ( w ; z ) (cid:1) x (cid:17) ≤ (cid:16) η (cid:107) S (cid:107) (cid:17) − sup x ∈ X D ( u x , w x ) , (4.12) for any u, w ∈ B η , and D defined in (4.1) . Proof.
First we show that B η is mapped to itself. For this let u ∈ B η bearbitrary. We start with the upper bound | Φ( u ; z ) | ≤ z + a + Su ) ≤ z ≤ η , where in the second inequality we employed the non-negativity property of S and that Im u ≥ . S Since | z | ≤ η − and η ≤ / (cid:107) a (cid:107) , we also find a lowerbound, | Φ( u ; z ) | ≥ | z | + | a | + | Su | ≥ η − + (cid:107) a (cid:107) + (cid:107) S (cid:107) η − ≥ η (cid:107) S (cid:107) . Now we use this as an input to establish the lower bound on the imaginary part,
Im Φ( u ; z ) = Im( z + a + Su ) | z + a + Su | ≥ | Φ( u ; z ) | Im z ≥ η (2 + (cid:107) S (cid:107) ) .
33e are left with establishing the inequality in (4.12). For that we use thethree properties of D in Lemma 4.1. By Property 1, the function D is invariantunder the isometries ζ (cid:55)→ − /ζ and ζ (cid:55)→ ζ − a x − Re z of H . Therefore for any u, w ∈ B η and x ∈ X : D (cid:16) (cid:0) Φ( u ; z ) (cid:1) x , (cid:0) Φ( w ; z ) (cid:1) x (cid:17) = D (cid:0) z + a x + ( Su ) x , z + a x + ( Sw ) x (cid:1) = D (cid:0) i Im z + ( Su ) x , i Im z + ( Sw ) x (cid:1) . (4.13)In case the non-negative functional S x ∈ B ∗ , defined through S x ( u ) := ( Su ) x ,vanishes identically, the expression in (4.13) vanishes as well. Thus we mayassume that S x (cid:54) = 0 . In view of Property 2 we estimate D (cid:0) i Im z + ( Su ) x , i Im z + ( Sw ) x (cid:1) ≤ (cid:16) z Im( Su ) x (cid:17) − (cid:16) z Im( Sw ) x (cid:17) − D (cid:0) ( Su ) x , ( Sw ) x (cid:1) . Plugging this back into (4.13) and recalling Im z ≥ η and (cid:107) Sw (cid:107) ≤ (cid:107) S (cid:107) η − , for z ∈ H η and w ∈ B η , respectively, we obtain D (cid:16)(cid:0) Φ( u ; z ) (cid:1) x , (cid:0) Φ( w ; z ) (cid:1) x (cid:17) ≤ (cid:16) η (cid:107) S (cid:107) (cid:17) − D (cid:0) ( Su ) x , ( Sw ) x (cid:1) . Using Property 3 in Lemma 4.1 we find D (cid:0) ( Su ) x , ( Sw ) x (cid:1) ≤ sup x ∈ X D ( u x , w x ) . This finishes the proof of the lemma.Lemma 4.2 shows that the sequence of iterates ( u ( n ) ) ∞ n =0 , with u ( n +1) :=Φ( u ( n ) ; z ) , is Cauchy for any initial function u (0) ∈ B η and any z ∈ H η .Therefore, ( u ( n ) ) n ∈ N converges to the unique fixed point m = m ( z ) ∈ B η of Φ( · ; z ) . We have therefore shown existence and uniqueness of (2.6) for anygiven z ∈ H η and thus, since η was arbitrary, even for all z ∈ H . In order to show that m x can be represented as a Stieltjes transform (cf. (2.8)),we will first prove that m x is a holomorphic function on H . We can use the sameargument as above on a space of function which are also z dependent. Namely,we consider the complete metric space, obtained by equipping the set B η := (cid:8) u : H η → B η : u is holomorphic (cid:9) , (4.14)of B η -valued functions u on H η , with the metric d η ( u , w ) := sup z ∈ H η d ( u ( z ) , w ( z )) , u , w ∈ B η . (4.15) 34ere the holomorphicity of u means that the map z (cid:55)→ φ ( u ( z )) is holomorphic on H η for any element φ in the dual space of B . Since the constant (1+ η / (cid:107) S (cid:107) ) − in (4.12) only depends on η , but not on z , we see that the function u (cid:55)→ Φ( u ) ,defined by (Φ( u ))( z ) := Φ( u ( z ); z ) , ∀ u ∈ B η , (4.16)inherits the contraction property from Φ( · ; z ) . Thus the iterates u ( n ) := Φ n ( u (0) ) for any initial function u (0) ∈ B η converge to the unique holomorphic func-tion m : H η → B η , which satisfies m ( z ) = (Φ( m ))( z ) for all z ∈ H η .Since η > was arbitrary and by the uniqueness of the solution on H η ,we see that there is a holomorphic function m : H → B + which satisfies m ( z ) = (Φ( m ))( z ) = Φ( m ( z ); z ) , for all z ∈ H . This function z (cid:55)→ m ( z ) isthe unique holomorphic solution of the QVE.Now we show the representation (2.8) for m ( z ) . We use that a holomorphicfunction φ : H → H on the complex upper half plane H is a Stieltjes transformof a probability measure on the real line if and only if | i η φ (i η ) + 1 | → as η → ∞ (cf. Theorem 3.5 in [Gar07]). In order to see that lim η →∞ sup x (cid:12)(cid:12) i η m x (i η ) + 1 (cid:12)(cid:12) = 0 , (4.17)we write the QVE in the form zm x ( z ) + 1 = − m x ( z ) ( a + Sm ( z )) x . We bound the right hand side by taking the uniform norms, | zm x ( z ) + 1 | ≤ (cid:107) a (cid:107)(cid:107) m ( z ) (cid:107) + (cid:107) S (cid:107)(cid:107) m ( z ) (cid:107) . We continue by using Im m ( z ) ≥ and the fact that S preserves positivity: | m ( z ) | = 1 | z + a + Sm ( z ) | ≤ z + a + Sm ( z )) ≤ z , ∀ z ∈ H . (4.18)Choosing z = i η , we get | i η m (i η ) + 1 | ≤ (cid:107) a (cid:107) η − + (cid:107) S (cid:107) η − , and hence (4.17) holds true. This completes the proof of the Stieltjes transformrepresentation (2.8).As the next step we show that the measures v x , x ∈ X , in (2.8) are supportedon an interval [ − Σ , Σ] , where Σ = (cid:107) a (cid:107) + 2 (cid:107) S (cid:107) / . We start by extending thesemeasures to functions on the complex upper-half plane. Definition 4.3 (Extended generating density) . Let m be the solution of theQVE. Then we define v x ( z ) := Im m x ( z ) , ∀ x ∈ X , z ∈ H . (4.19) 35he union of the supports of the generating measures (2.8) on the real line isdenoted by: supp v := (cid:91) x ∈ X supp v x | R . (4.20)This extension is consistent with the generating measure v x appearing in(2.8) since v x ( z ) , z ∈ H , is obtained by regularizing the generating measurewith the Cauchy-density at the scale η > . Indeed, (4.19) is equivalent to v x ( τ + i η ) = (cid:90) ∞−∞ η Π (cid:16) τ − ωη (cid:17) v x (d ω ) , Π( λ ) := 1 π
11 + λ , (4.21)for any τ ∈ R and η > .We will now show that the support of the generating measure v lies insidean interval with endpoints ± Σ , with Σ = (cid:107) a (cid:107) + 2 (cid:107) S (cid:107) / . To this end, supposethat (cid:107) m ( z ) (cid:107) < | z | − (cid:107) a (cid:107) (cid:107) S (cid:107) , for some | z | > Σ , (4.22)where we have used (cid:107) S (cid:107) > . Feeding (4.22) into the QVE we obtain a slightlybetter bound: (cid:107) m ( z ) (cid:107) ≤ | z | − (cid:107) a (cid:107) − (cid:107) S (cid:107)(cid:107) m ( z ) (cid:107) ≤ | z | − (cid:107) a (cid:107) . Denoting D ε := (cid:110) z ∈ H : | z | ≥ (cid:107) a (cid:107) + 2 (cid:107) S (cid:107) / (1 + ε ) (cid:111) , for an arbitrary ε ∈ (0 , / , we have shown that the range of the restriction ofthe norm function (cid:107) m (cid:107) to D ε is a union of two disjoint sets, i.e., z (cid:55)→ (cid:107) S (cid:107) / (cid:107) m ( z ) (cid:107) : D ε → (cid:2) , (1 + ε ) − (cid:3) ∪ (cid:2) ε, ∞ (cid:1) . (4.23)From the Stieltjes transform representation (2.8) we see that (4.23) is a contin-uous function. The bound (4.17) implies (cid:107) m (i η ) (cid:107) ≤ (1 + ε ) − for sufficientlylarge η > . For large η we also have i η ∈ D ε . As D ε is a connected set, thecontinuity of (4.23) implies that for any ε > (cid:107) S (cid:107) / (cid:107) m ( z ) (cid:107) ≤ (1 + ε ) − , when | z | ≥ Σ + 2 (cid:107) S (cid:107) / ε . (4.24)Now we take the imaginary part of the QVE to get v ( z ) | m ( z ) | = − Im 1 m ( z ) = Im z + Sv ( z ) , (4.25)where v ( z ) is from (4.19). Taking the norms in this formula and rearranging it,we obtain (cid:16) − (cid:0) (cid:107) S (cid:107) / (cid:107) m ( z ) (cid:107) (cid:1) (cid:17) (cid:107) v ( z ) (cid:107) ≤ (cid:107) m ( z ) (cid:107) Im z . (4.26) 36onsider z := τ + i η , with | τ | > Σ and η > . Then the coefficient in front of (cid:107) v ( z ) (cid:107) is larger than (1 − (1 + ε ) − ) > , with ε := ( | τ | − Σ) / (2 (cid:107) S (cid:107) / ) > . Inparticular, this bound is uniform in η . We estimate (cid:107) m (cid:107) on the right hand sideof (4.26) by (4.24). Thus we see that v ( τ + i η ) → by taking the limit η → locally uniformly for | τ | > Σ . F and structural L -bound In this section we finish the proof of Theorem 2.1 by considering the remainingthe special case a = 0 . First we note that the real and imaginary parts of thesolution m of the QVE are odd and even functions of Re z with fixed Im z ,respectively when a = 0 , i.e., m ( − z ) = − m ( z ) , ∀ z ∈ H . (4.27)Combining this with (4.20) we obtain the symmetry of the generating measure.The proof of the upper bound (2.9) on the L -norm of m ( z ) relies on the anal-ysis of the following symmetric positivity preserving operator F ( z ) , generatedby m ( z ) . Definition 4.4 (Operator F ) . The operator F ( z ) : B → B for z ∈ H , isdefined by F ( z ) w := | m ( z ) | S ( | m ( z ) | w ) , w ∈ B , (4.28)where m ( z ) is the solution of the QVE at z .The operator F ( z ) will play a central role in the upcoming analysis. Inparticular, using F ( z ) we prove the structural L -bound for the solution. Lemma 4.5 (Structural L -bound) . Assuming A1 , we have (cid:107) m ( z ) (cid:107) ≤ z, { a x : x ∈ X } ) , ∀ z ∈ H . (4.29) Proof.
We start by writing the QVE in the form − ( z + a ) m ( z ) = 1 + m ( z ) Sm ( z ) . (4.30)Taking the L -norm on both sides yields (cid:107) m ( z ) (cid:107) ≤ (cid:0) (cid:107) m ( z ) Sm ( z ) (cid:107) (cid:1) (cid:107) ( z + a ) − (cid:107) ≤ (cid:107) F ( z ) (cid:107) → dist( z, { a x : x ∈ X } ) . (4.31)Here the last bound follows by writing | m ( z ) Sm ( z ) | ≤ | m ( z ) | S | m ( z ) | = F ( z ) e ,where e ∈ B stands for the constant function equal to one, and then estimating: (cid:107) m ( z ) Sm ( z ) (cid:107) = (cid:107) F ( z ) e (cid:107) ≤ (cid:107) F ( z ) (cid:107) L → L . (4.32) 37he bound (4.29) now follows by bounding F ( z ) as an operator on L . In fact,we now show that (cid:107) F ( z ) (cid:107) L → L < , ∀ z ∈ H . (4.33)The operator S is bounded on B . Therefore (cid:107) S (cid:107) L ∞ → L ∞ ≤ (cid:107) S (cid:107) < ∞ . Since S is symmetric we have (cid:107) S (cid:107) L → L = (cid:107) S (cid:107) L ∞ → L ∞ . Using the Riesz-Thorin inter-polation theorem we hence see that (cid:107) S (cid:107) L p → L p ≤ (cid:107) S (cid:107) , for every p ∈ [1 , ∞ ] .For each z ∈ H the operator F ( z ) is also bounded on L q , as | m ( z ) | is triviallybounded by (Im z ) − (cf. (4.18)). Furthermore, from the Stieltjes transformrepresentation (2.8) it follows that m x ( z ) is also bounded away from zero: Im m x ( z ) ≥ π Im z (Σ + | z | ) , ∀ x ∈ X . (4.34)The estimate (4.33) is obtained by considering the imaginary part (4.25) ofthe QVE. Rewriting this equation in terms of F = F ( z ) we get v | m | = | m | Im z + F v | m | . (4.35)In order to avoid excess clutter we have suppressed the dependence of z in ournotation. The trivial lower bound (4.34) on v ( z ) and the trivial upper bound | m ( z ) | ≤ (Im z ) − imply that there is a scalar function ε : H → (0 , , such that F v | m | ≤ (1 − ε ) v | m | , ε := (Im z ) inf x | m x | v x ∈ (0 , . (4.36)The fact that ε ∈ (0 , follows from (4.35), the strict pointwise positivity of v and the positivity preserving property of F . If ε = 1 we have nothing to showsince F = 0 in this case. If ε < , then we apply Lemma 4.6 below with thechoices, T := F − ε , and h := v | m | (cid:38) (Im z ) | z | , to conclude (cid:107) F (cid:107) L → L < . Lemma 4.6 (Subcontraction) . Let T be a bounded symmetric operator on L that preserves non-negative functions, i.e., if u ≥ almost everywhere, thenalso T u ≥ almost everywhere. If there exists an almost everywhere positivefunction h ∈ L , such that almost everywhere T h ≤ h , then (cid:107) T (cid:107) L → L ≤ . The proof of Lemma 4.6 is postponed to Appendix A.138 hapter 5
Properties of solution
In this chapter we prove various technical estimates for the solution m of theQVE and the associated operator F (cf. (4.4)). In the second half of the chapterwe start analyzing the stability of the QVE under small perturbations. For thestability analysis, we introduce the concept of the (spectral) gap of an operator. Definition 5.1 (Spectral gap) . Let T : L → L be a compact self-adjointoperator. The spectral gap Gap( T ) is the difference between the two largesteigenvalues of | T | . If (cid:107) T (cid:107) L → L is a degenerate eigenvalue of | T | then Gap( T ) =0 . We will frequently use comparison relations ∼ , (cid:46) in the sequel that dependon a certain set of model parameters (c.f. Convention 2.3). This set may bedifferent in various lemmas and propositions. In order to avoid constantly listingthem we extend Convention 2.3 as follows: Convention 5.2 (Standard model parameters) . The norm (cid:107) a (cid:107) is always consid-ered a model parameter. If the property A2 of S is assumed in some statement,then the associated constant (cid:107) S (cid:107) L → B is automatically a model parameter. Sim-ilarly, if A3 is assumed, then ρ and L are considered model parameters. Anyadditional model parameters will be declared explicitly. Naturally, inside a proofof a statement the comparison relations depend on the model parameters of thatstatement. m and F The following proposition collects the most important estimates in the specialcase when the solution is uniformly bounded.
Proposition 5.3 (Estimates when solution is bounded) . Suppose S satisfies A1-3 . Additionally, assume that for some I ⊆ R , and Φ < ∞ the uniformbound ||| m ||| I ≤ Φ , pplies. Then, considering Φ an additional model parameter, the following es-timates apply for every z ∈ H , with Re z ∈ I : (i) The solution m of the QVE satisfies the bounds | m x ( z ) | ∼
11 + | z | , ∀ x ∈ X . (5.1)(ii) The imaginary part is comparable to its average, i.e. v x ( z ) ∼ (cid:104) v ( z ) (cid:105) , ∀ x ∈ X . (5.2)(iii) The largest eigenvalue λ ( z ) of F ( z ) is single, and satisfies λ ( z ) ≤ , and λ ( z ) = (cid:107) F ( z ) (cid:107) L → L ∼
11 + | z | . (5.3)(iv) The operator F ( z ) has a uniform spectral gap, i.e., Gap( F ( z )) ∼ (cid:107) F ( z ) (cid:107) L → L . (5.4)(v) The unique eigenvector f ( z ) ∈ B , satisfying F ( z ) f ( z ) = λ ( z ) f ( z ) , f x ( z ) ≥ , and (cid:107) f ( z ) (cid:107) = 1 , (5.5) is comparable to , i.e. f x ( z ) ∼ , ∀ x ∈ X . (5.6)For a complete proof of Proposition 5.3 (cf. p. 47) we first prove variousauxiliary results, under the standing assumption in this chapter: • S satisfies A1-3 .We start by pointing out a few simple properties of S that we need in thefollowing. The smoothing condition A2 implies that for every x ∈ X the linearfunctional S x : L → R , w (cid:55)→ ( Sw ) x is bounded. Hence, the row-function y (cid:55)→ S xy is in L . The family of functions satisfies sup x (cid:107) S x (cid:107) = (cid:107) S (cid:107) L → B . Thebound (2.12) implies that S is a Hilbert-Schmidt operator.The uniform primitivity condition A3 guarantees that norms of the rowfunctions S x as well as various operator norms of S are comparable to one.Indeed, letting x ∈ X be fixed and choosing the constant function u = 1 in(2.11), we obtain ρ ≤ (cid:90) ( S L ) xy π (d y ) ≤ (cid:18)(cid:90) S xu π (d u ) (cid:19)(cid:18) sup t (cid:90) ( S L − ) ty π (d y ) (cid:19) ≤ (cid:107) S L − (cid:107)(cid:104) S x (cid:105) . (cid:107) S L − (cid:107) ≤ (cid:107) S (cid:107) L − this yields the first inequality of ρ (cid:107) S (cid:107) − ( L − ≤ (cid:104) S x (cid:105) ≤ (cid:107) S (cid:107) , x ∈ X . (5.7)The last bound is trivial since (cid:107) S (cid:107) = sup x (cid:104) S x (cid:105) . By Riesz-Thorin interpolationtheorem (cf. proof of Lemma 4.5) we have (cid:107) S (cid:107) L p → L p ≤ (cid:107) S (cid:107) . On the other hand,letting S act on the constant function, we have (cid:107) S (cid:107) L p → L p ≥ inf x (cid:104) S x (cid:105) . Combining this with (5.7), the trivial bound (cid:107) S (cid:107) ≤ (cid:107) S (cid:107) L → B , and the fact that (cid:107) S (cid:107) L → B is a model parameter (cf. Convention 5.2), we thus conclude (cid:104) S x (cid:105) ∼ , (cid:107) S (cid:107) ∼ , and (cid:107) S (cid:107) L p → L p ∼ , p ∈ [ 1 , ∞ ] . (5.8)The following lemma shows that a component | m x ( z ) | may diverge onlywhen (cid:107) m ( z ) (cid:107) = ∞ or (cid:104) v ( z ) (cid:105) = 0 . Furthermore, the lemma implies that if acomponent | m x ( z ) | , for some x ∈ X , approaches zero while z stays bounded,then another component | m y ( z ) | will always diverge at the same time. Lemma 5.4 (Constraints on solution) . If S satisfies A1-3 , then: (i)
The solution m of the QVE satisfies for every x ∈ X and z ∈ H : min (cid:26)
11 + | z | , inf y | z − a y | + 1 (cid:107) m ( z ) (cid:107) (cid:27) (cid:46) (cid:12)(cid:12) m x ( z ) (cid:12)(cid:12) (cid:46) min (cid:26) y | m y ( z ) | L − (cid:104) v ( z ) (cid:105) , z, supp v ) (cid:27) . (5.9)(ii) The imaginary part, v x ( z ) is comparable to its average, such that for every x ∈ X and z ∈ H with | z | ≤ : inf y (cid:12)(cid:12) m y ( z ) (cid:12)(cid:12) L (cid:46) v x ( z ) (cid:104) v ( z ) (cid:105) (cid:46) (cid:18) y | m y ( z ) | (cid:19) (cid:107) m ( z ) (cid:107) . (5.10) For | z | ≥ the function v satisfies v x ( z ) ∼ (cid:104) v ( z ) (cid:105) . These bounds simplify considerably when m = m ( z ) is uniformly boundedfor every z (cf. Proposition 5.3). Proof.
We start by proving the lower bound on | m | . This is done by estab-lishing an upper bound on / | m | . Using the QVE we find | m | = | z + a + Sm | ≤ | z | + (cid:107) a (cid:107) + (cid:107) S (cid:107) L → B (cid:107) m (cid:107) (cid:46) | z | + (cid:107) m (cid:107) . (5.11)Taking the reciprocal on both sides yields | m | (cid:38) min (cid:8) (1 + | z | ) − , (cid:107) m (cid:107) − (cid:9) .Combining the L -norm with (4.29) yields the lower bound in (5.9).41ow we will prove the upper bound on | m | . To this end, recall that m x ( z ) = 1 π (cid:90) R v x (d τ ) τ − z , where v x /π is a probability measure. Bounding the denominator from below by dist( z, supp v ) , with supp v = ∪ x supp v x , we obtain one of the upper bounds of(5.9): | m x ( z ) | ≤ z, supp v ) . For the derivation of the second upper bound we rely on the positivity ofthe imaginary part of m : | m | = 1 | Im ( z + a + Sm ) | ≤ Sv . (5.12)In order to continue we will now bound Sv from below. This is achieved byestimating v from below by (cid:104) v (cid:105) . Indeed, writing the imaginary part of theQVE, as v | m | = − Im 1 m = Im z + Sv , and ignoring Im z > , yields v ≥ | m | Sv ≥ φ Sv , (5.13)where we introduced the abbreviation φ := inf x | m x | . Now we make use of the uniform primitivity A3 of S and of (5.13). In this waywe get the lower bound on Sv , Sv ≥ φ S v ≥ . . . ≥ φ L − S L v ≥ φ L − ρ (cid:104) v (cid:105) , Plugging this back into (5.12) finishes the proof of the upper bound on | m | .We continue by showing the claim concerning v/ (cid:104) v (cid:105) . We start with the lowerbound. We use (5.13) in an iterative fashion and employ assumption A3 , v ≥ φ Sv ≥ . . . ≥ φ L S L v ≥ φ L ρ (cid:104) v (cid:105) . (5.14)This proves the lower bound v/ (cid:104) v (cid:105) (cid:38) φ L .In order to derive upper bounds for the ratio v/ (cid:104) v (cid:105) , we first write v = | m | (Im z + Sv ) ≤ (cid:107) m (cid:107) (Im z + Sv ) . (5.15)We will now bound Im z and Sv in terms of (cid:104) v (cid:105) . We start with Im z . Bydropping the term Sv from (5.15), and estimating | m | ≥ φ , we get v ≥ φ Im z .Averaging this yields Im z ≤ (cid:104) v (cid:105) φ . (5.16) 42n order to bound Sv , we apply S on both sides of (5.15), and use the boundon Im z , to get Sv ≤ (cid:18) (cid:104) v (cid:105) φ + S v (cid:19) (cid:107) m (cid:107) . (5.17)The expression involving S is useful, as we may now estimate the kernel ( S ) xy uniformly: ( S ) xy ≤ (cid:104) S x , S y (cid:105) ≤ (cid:107) S x (cid:107) (cid:107) S y (cid:107) ≤ sup x (cid:107) S x (cid:107) = (cid:107) S (cid:107) → B ∼ . (5.18)In particular, S v ≤ (cid:107) S (cid:107) → B (cid:104) v (cid:105) ∼ (cid:104) v (cid:105) , and thus Sv (cid:46) (cid:16) φ (cid:17) (cid:107) m (cid:107) (cid:104) v (cid:105) . With this and (5.16) plugged back into (5.15) we get the upper bound of (5.10): v (cid:46) (cid:16) φ (cid:17) (cid:107) m (cid:107) (cid:104) v (cid:105) . Here we have also used the lower bound (cid:107) m ( z ) (cid:107) (cid:38) to replace (cid:107) m (cid:107) by (cid:107) m (cid:107) inthe regime | z | ≤ , where Σ = (cid:107) a (cid:107) + 2 (cid:107) S (cid:107) / ∼ by (5.8). The lower boundon (cid:107) m (cid:107) follows directly from the QVE and (cid:107) S (cid:107) ∼ : | ( z + a + Sm ) m | (cid:46) (cid:0) | z | + (cid:107) a (cid:107) + (cid:107) S (cid:107)(cid:107) m (cid:107) (cid:1) (cid:107) m (cid:107) . On the other hand, if | z | ≥ , then v ( z ) ∼ (cid:104) v ( z ) (cid:105) holds because v x ( z ) is theharmonic extension (2.8) of the measure v x (d τ ) which is supported inside theinterval with endpoints ± Σ .Since the solution m ( z ) for z ∈ H of the QVE is bounded by the trivial bound(cf. (4.18)), the operator F ( z ) introduced in Definition 4.4 is a Hilbert-Schmidtoperator. Consistent with the notation for S we write F xy ( z ) for the symmetricnon-negative measurable kernel representing this operator. The largest eigen-value and the corresponding eigenvector of F ( z ) will play a key role when weanalyze the sensitivity of m ( z ) to changes in z , or more generally, to any per-turbations of the QVE. The following lemma provides an exact formula for thiseigenvalue. Lemma 5.5 (Operator F ) . Assume that S satisfies A1-3 . Then for every z ∈ H the operator F ( z ) , defined in (4.28) , is a Hilbert-Schmidt integral operator on L , with the integral kernel F xy ( z ) = | m x ( z ) | S xy | m y ( z ) | . (5.19) The norm λ ( z ) := (cid:107) F ( z ) (cid:107) L → L is a single eigenvalue of F ( z ) , and it satisfies: (cid:107) F ( z ) (cid:107) L → L = 1 − Im zα ( z ) (cid:10) f ( z ) | m ( z ) | (cid:11) < , z ∈ H . (5.20) 43 ere the positive eigenvector f : H → B is defined by (5.5) , while α : H → (0 , ∞ ) is the size of the projection of v/ | m | onto the direction f : α ( z ) := (cid:68) f ( z ) , v ( z ) | m ( z ) | (cid:69) . (5.21) Proof.
The existence and uniqueness of (cid:107) F ( z ) (cid:107) L → L as a non-degenerate eigen-value and f ( z ) as the corresponding eigenvector satisfying (5.5) follow fromLemma 5.6 below by choosing r := | m ( z ) | , using the trivial bound (cid:107) m ( z ) (cid:107) (cid:46) (Im z ) − to argue (using (5.9)) that also r − := inf x | m x | > .In order to obtain (5.20) we take the inner product of (4.35) with f = f ( z ) .Since F ( z ) is symmetric, we find (cid:68) f v | m | (cid:69) = (cid:10) f | m | (cid:11) Im z + (cid:107) F (cid:107) L → L (cid:68) f v | m | (cid:69) . (5.22)Rearranging the terms yields the identity (5.20).The following lemma demonstrates how the spectral gap, Gap( F ( z )) , thenorm and the associated eigenvector of F ( z ) depend on the component wiseestimates of | m x ( z ) | . Since we will later need this result for a general positivefunction r : X → (0 , ∞ ) in the role of | m ( z ) | we state the result for a generaloperator (cid:98) F ( r ) below. Lemma 5.6 (Maximal eigenvalue of scaled S ) . Assume S satisfies A1-3 . Con-sider an integral operator (cid:98) F ( r ) : L → L , parametrized by r ∈ B , with r x ≥ for each x , and defined through the integral kernel (cid:98) F xy ( r ) := r r S xy r y . (5.23) If there exist upper and lower bounds, < r − ≤ r + < ∞ , such that r − ≤ r x ≤ r + , ∀ x ∈ X , then (cid:98) F ( r ) is Hilbert-Schmidt, and (cid:98) λ ( r ) := (cid:107) (cid:98) F ( r ) (cid:107) L → L is a single eigenvaluesatisfying the upper and lower bounds r − (cid:46) (cid:98) λ ( r ) (cid:46) r . (5.24) Furthermore, there is a spectral gap,
Gap( (cid:98) F ( r )) (cid:38) r L − r − (cid:98) λ ( r ) − L +5 , (5.25) and the unique eigenvector, (cid:98) f ( r ) ∈ L , satisfying (cid:98) F ( r ) (cid:98) f ( r ) = (cid:98) λ ( r ) (cid:98) f ( r ) , (cid:98) f x ( r ) ≥ , and (cid:107) (cid:98) f ( r ) (cid:107) = 1 , (5.26) 44 s comparable to its average in the sense that (cid:18) r − (cid:98) λ ( r ) (cid:19) L (cid:46) (cid:98) f x ( r ) (cid:104) (cid:98) f ( r ) (cid:105) (cid:46) r (cid:98) λ ( r ) . (5.27) If (cid:98) F is interpreted as a bounded operator on B , then the following relationshipbetween the norm of the L -resolvent and the B -resolvent holds (cid:107) ( (cid:98) F ( r ) − ζ ) − (cid:107) (cid:46) | ζ | (cid:18) r (cid:107) ( (cid:98) F ( r ) − ζ ) − (cid:107) L → L (cid:19) , (5.28) for every ζ (cid:54)∈ Spec( (cid:98) F ( r )) ∪ { } . Feeding (5.24) into (5.27) yields Φ − L (cid:104) (cid:98) f ( r ) (cid:105) (cid:46) (cid:98) f ( r ) (cid:46) Φ (cid:104) (cid:98) f ( r ) (cid:105) , where Φ := r + /r − . For the proof of Lemma 5.6 we need a simple on the spectral gap thatis well known in various forms. For the convenience of the reader we include aproof in Appendix A.2. Lemma 5.7 (Spectral gap for positive bounded operators) . Let T be a sym-metric compact integral operator on L ( X ) with a non-negative integral kernel T xy = T yx ≥ . Then Gap( T ) ≥ (cid:18) (cid:107) h (cid:107) L (cid:107) h (cid:107) (cid:19) inf x,y ∈ X T xy , where h is an eigenfunction with T h = (cid:107) T (cid:107) L → L h . Proof of Lemma 5.6.
Since S is compact, and r ≤ r + also (cid:98) F = (cid:98) F ( r ) iscompact. The operator (cid:98) F preserves the cone of non-negative functions u ≥ . Hence by the Krein-Rutman theorem (cid:98) λ = (cid:107) (cid:98) F (cid:107) L → L is an eigenvalue, andthere exists a non-negative normalized eigenfunction (cid:98) f ∈ L ( X ) correspondingto (cid:98) λ . The smoothing property A2 and the uniform primitivity assumption A3 combine to inf x,y ∈ X ( S L ) xy ≥ ρ . Since r − > , it follows that the integral kernel of (cid:98) F L is also strictly positiveeverywhere. In particular, (cid:98) F is irreducible, and thus the eigenfunction (cid:98) f isunique.Now we derive the upper bound for (cid:98) λ . Since (cid:107) w (cid:107) p ≤ (cid:107) w (cid:107) q , for p ≤ q , weobtain (cid:98) λ = (cid:107) (cid:98) F (cid:107) → L = (cid:107) (cid:98) F (cid:107) L → L ≤ (cid:107) (cid:98) F (cid:107) L → B = sup x,y ( (cid:98) F ) xy ≤ r (cid:107) S (cid:107) → B , which implies (cid:98) λ (cid:46) r . Here we have used ( S ) xy = (cid:104) S x , S y (cid:105) ≤ (cid:107) S x (cid:107) (cid:107) S y (cid:107) , and sup x (cid:107) S x (cid:107) = (cid:107) S (cid:107) L → B to estimate: ( (cid:98) F ) xy ≤ r ( S ) xy ≤ r (cid:107) S (cid:107) → B . (5.29) 45or the lower bound on (cid:98) λ , we use first (5.8) and (5.8) to get (cid:82)(cid:82) π (d x ) π (d y ) S xy ∼ . Therefore (cid:98) λ = (cid:107) (cid:98) F (cid:107) L → L ≥ (cid:104) e, (cid:98) F e (cid:105) ≥ r − (cid:90) (cid:90) π (d x ) π (d y ) S xy ∼ r − , (5.30)where e ∈ B is a function equal to one e x = 1 .Now we show the upper bound for the eigenvector. Applying (5.29), and (cid:104) (cid:98) f (cid:105) = (cid:107) (cid:98) f (cid:107) ≤ (cid:107) (cid:98) f (cid:107) = 1 , yields (cid:98) λ (cid:98) f x = ( (cid:98) F (cid:98) f ) x (cid:46) r (cid:104) (cid:98) f (cid:105) ≤ r . This shows the upper bound on (cid:98) f x / (cid:104) (cid:98) f (cid:105) and, in addition, (cid:98) f x (cid:46) r / (cid:98) λ .In order to estimate the ratios (cid:98) f x / (cid:104) (cid:98) f (cid:105) , x ∈ X , from below, we consider theoperator T := (cid:16) (cid:98) F (cid:98) λ (cid:17) L . (5.31)Using inf x,y ( S L ) xy ≥ ρ , we get inf x,y T xy ≥ r L − (cid:98) λ L ( S L ) xy (cid:38) (cid:16) r − (cid:98) λ (cid:17) L . Hence, we find a lower bound on (cid:98) f through (cid:98) f x = ( T (cid:98) f ) x (cid:38) (cid:16) r − (cid:98) λ (cid:17) L (cid:104) (cid:98) f (cid:105) . (5.32)In order to prove (5.25), we apply Lemma 5.7 to the operator T , to get Gap( T ) ≥ inf x,y T xy (cid:107) (cid:98) f (cid:107) (cid:38) ( r − / (cid:98) λ ) L ( r / (cid:98) λ ) = r L − r − (cid:98) λ − ( L − . Since L ∼ , this implies, Gap( (cid:98) F ) (cid:98) λ = 1 − (cid:0) − Gap( T ) (cid:1) /L ≥ Gap( T ) L ∼ r L − r − (cid:98) λ − ( L − . Finally, we show the bound (5.28). Here the smoothing condition A2 on S is crucial. Let d, w ∈ B satisfy ( (cid:98) F − ζ ) − w = d . For ζ / ∈ Spec( (cid:98) F ) ∪ { } , wehave (cid:107) d (cid:107) ≤ (cid:107) ( (cid:98) F − ζ ) − (cid:107) L → L (cid:107) w (cid:107) ≤ (cid:107) ( (cid:98) F − ζ ) − (cid:107) L → L (cid:107) w (cid:107) . (5.33)Now, using (cid:107) S (cid:107) L → B (cid:46) , we bound the uniform norm of d from above by thecorresponding L -norm: | ζ | (cid:107) d (cid:107) = (cid:107) (cid:98) F d − w (cid:107) ≤ (cid:107) (cid:98) F (cid:107) L → B (cid:107) d (cid:107) + (cid:107) w (cid:107) ≤ r (cid:107) S (cid:107) L → B (cid:107) d (cid:107) + (cid:107) w (cid:107) . The estimate (5.28) now follows by using the operator norm on L for the resol-vent, i.e., the inequality (5.33) to estimate (cid:107) d (cid:107) by (cid:107) w (cid:107) .46 roof of Proposition 5.3. All the claims follow by combining Lemma 5.4,Lemma 5.5 and Lemma 5.6. Indeed, let z ∈ I +i (0 , ∞ ) , so that (cid:107) m ( z ) (cid:107) ≤ Φ ∼ .Since supp v ⊂ [ − Σ , Σ] , with Σ = (cid:107) a (cid:107) +2 (cid:107) S (cid:107) / ∼ (cf. (5.8)), the upper boundof (5.9) yields (cid:107) m ( z ) (cid:107) ≤ | z | − for | z | ≥ . Thus (cid:107) m ( z ) (cid:107) (cid:46) (1 + | z | ) − for all Re z ∈ I . Using this upper bound in the first estimate of (5.9) yields the part(i) of the proposition: | m x ( z ) | ∼ (1 + | z | ) − , x ∈ X , Re z ∈ I . (5.34)When | z | ≤ the comparison relation v x ( z ) ∼ (cid:104) v ( z ) (cid:105) follows by plugging(5.34) into (5.10). If | z | > , then v and its average are comparable due to theStieltjes transform representation (2.8) and the bound (2.7) for the support of v | R . This completes the proof of the part (ii).For the claims concerning the operator F ( z ) we use the formula (5.19) toidentify F ( z ) = (cid:98) F ( | m ( z ) | ) , where (cid:98) F ( r ) for r ∈ B satisfying r ≥ , is theoperator from Lemma 5.6.The parts (iii-v) follow from Lemma 5.6 with the choice r − := inf x | m x | and r + := sup x | m x | , since r ± ∼ (1 + | z | ) − by (5.34). B The next lemma introduces the operator B that plays a central role in thestability analysis of the QVE. At the end of this section (Lemma 5.11) we presentthe first stability result for the QVE which is effective when m is uniformlybounded and B − is bounded as operator on B . Subtracting the QVE from(5.35) an elementary algebra yields the following lemma. Lemma 5.8 (Perturbations) . Suppose g, d ∈ B , with inf x | g x | > , satisfy theperturbed QVE, − g = z + a + Sg + d , (5.35) at some fixed z ∈ H and suppose m = m ( z ) solves the unperturbed QVE. Then u := g − m ( z ) | m ( z ) | , (5.36) satisfies the equation B u = e − i q u F u + | m | d + | m | e − i q ud , (5.37) where the operator B = B ( z ) , and the function q = q ( z ) : X → [0 , π ) are givenby B := e − i2 q − F , and e i q := m | m | . (5.38) 47emma 5.8 shows that the inverse of the non-selfadjoint operator B ( z ) playsan important role in the stability of the QVE against perturbations. In the nextlemma we estimate the size of this operator in terms of the solution of the QVE. Lemma 5.9 (Bounds on B − ) . Assume
A1-3 , and consider z ∈ H such that | z | ≤ . Then the following estimates hold: (i) If (cid:107) m ( z ) (cid:107) ≤ Λ , for some Λ < ∞ , then (cid:107) B ( z ) − (cid:107) L → L (cid:46) (cid:104) v ( z ) (cid:105) − , and (cid:107) B ( z ) − (cid:107) (cid:46) (cid:104) v ( z ) (cid:105) − , (5.39) with Λ considered an additional model parameter. (ii) If (cid:107) m ( z ) (cid:107) ≤ Φ , for some Φ < ∞ , then (cid:107) B ( z ) − (cid:107) (cid:46) (cid:107) B ( z ) − (cid:107) L → L (5.40a) (cid:46) ( | σ ( z ) | + (cid:104) v ( z ) (cid:105) ) − (cid:104) v ( z ) (cid:105) − , (5.40b) with the function σ : H → R , defined by σ ( z ) := (cid:10) f ( z ) sign Re m ( z ) (cid:11) , (5.41) and Φ considered an additional model parameter. We remark that (5.40b) improves on the analogous bound (cid:107) B − (cid:107) (cid:46) (cid:104) v (cid:105) − that was proven in [AEK16b]. We will see below that (5.40b) is sharp in termsof powers of (cid:104) v (cid:105) . On the other hand, the exponents in (5.39) may be improved.For the proof of Lemma 5.9 we need the following auxiliary result which wasprovided as Lemma 5.8 in [AEK16b]. Since it plays a fundamental role in theanalysis its proof is reproduced in Appendix A.2. Lemma 5.10 (Norm of B − -type operators on L ) . Let T be a compact self-adjoint and U a unitary operator on L ( X ) . Suppose that Gap( T ) > and (cid:107) T (cid:107) L → L ≤ . Then there exists a universal positive constant C such that (cid:107) ( U − T ) − (cid:107) L → L ≤ C Gap( T ) | − (cid:107) T (cid:107) L → L (cid:104) h, U h (cid:105)| , (5.42) where h is the L -normalized eigenvector of T , corresponding to the non-degenerateeigenvalue (cid:107) T (cid:107) L → L . Proof of Lemma 5.9.
We will prove the estimates (5.39) and (5.40) partlyin parallel. Depending on the case, z is always assumed to lie inside the appro-priate domain, i.e., either z is fixed such that (cid:107) m ( z ) (cid:107) ≤ Λ , or ||| m ||| { τ } ≤ Φ ,with Re z = τ . Besides this, we consider z to be fixed. Correspondingly, thecomparison relations in this proof depend on either ( ρ, L, (cid:107) a (cid:107) , (cid:107) S (cid:107) L → B , Λ) or ( ρ, L, (cid:107) a (cid:107) , (cid:107) S (cid:107) L → B , Φ) (cf. Convention 5.2). We will also drop the explicit z -arguments in order to make the following formulas more transparent. In bothcases the lower bound | m x ( z ) | (cid:38) follows from (5.9).48e start the analysis by noting that it suffices to consider only the norm of B − on L , since (cid:107) B − (cid:107) (cid:46) (cid:107) m (cid:107) (cid:107) B − (cid:107) L → L . (5.43)In order to see this, we use the smoothing property A2 of S as in the proof of(5.28) before. In fact, besides replacing the complex number ζ with the function e q , the proof of (5.33) carries over without further changes.By the general property (5.20) of F we know that (cid:107) F (cid:107) L → L ≤ . Fur-thermore, it is immanent from the definition of F and (5.7) that (cid:107) F (cid:107) L → L (cid:38) inf x | m x | (cid:38) in both of the considered cases. This shows that the hypothesesof Lemma 5.10 are met, and hence (cid:107) B − (cid:107) L → L (cid:46) Gap( F ) − (cid:12)(cid:12) − (cid:107) F (cid:107) L → L (cid:104) e i2 q f (cid:105) (cid:12)(cid:12) − , (5.44)where we have also used (cid:107) F (cid:107) L → L ∼ . Now, by basic trigonometry, (cid:104) e i2 q f (cid:105) = (cid:10) (1 − q ) f (cid:11) + i2 (cid:10) f sin q cos q (cid:11) , and therefore we get (cid:12)(cid:12) − (cid:107) F (cid:107) L → L (cid:10) e i2 q f (cid:11)(cid:12)(cid:12) (cid:38) − (cid:107) F (cid:107) L → L + (cid:107) f sin q (cid:107) + (cid:12)(cid:12)(cid:10) f sin q cos q (cid:11)(cid:12)(cid:12) . (5.45)Here, we have again used (cid:46) (cid:107) F (cid:107) L → L ≤ . Substituting this back into (5.44)yields (cid:107) B − (cid:107) L → L ≤ F ) 11 − (cid:107) F (cid:107) L → L + (cid:107) f sin q (cid:107) + |(cid:104) f sin q cos q (cid:105)| . (5.46) Case 1 ( m with L -bound): In this case we drop the (cid:104) f sin q cos q (cid:105) term andestimate (cid:107) f sin q (cid:107) ≥ (cid:107) f (cid:107) inf x sin q x = inf x v x | m x | (cid:38) (cid:104) v (cid:105) , (5.47)where the bounds (cid:107) m (cid:107) (cid:46) Λ C (cid:104) v (cid:105) − ∼ (cid:104) v (cid:105) − and v (cid:38) Λ − C (cid:104) v (cid:105) ∼ (cid:104) v (cid:105) fromLemma 5.4 were used in the last inequality. Plugging (5.47) back into (5.46),and using (5.25) to estimate Gap( F ) = Gap( (cid:98) F ( | m | )) (cid:38) Λ − C (cid:107) m (cid:107) − (cid:38) (cid:104) v (cid:105) yields the desired bound: (cid:107) B − (cid:107) L → L (cid:46) Gap( F ) − (cid:107) f sin q (cid:107) − (cid:46) (cid:104) v (cid:105) − (cid:104) v (cid:105) − ∼ (cid:104) v (cid:105) − . (5.48)The operator norm bound on B follows by combining this estimate with (5.43),and then using (5.9) to estimate (cid:107) m (cid:107) (cid:46) Λ − L +2 (cid:104) v (cid:105) − ∼ (cid:104) v (cid:105) − . Case 2 ( m uniformly bounded): Now we assume (cid:107) m (cid:107) ≤ Φ ∼ , and thus allthe bounds of Proposition 5.3 are at our disposal. This will allow us to extract49seful information from the term |(cid:104) f sin q cos q (cid:105)| in (5.46) that was neglected inthe derivation of (5.48). Clearly, |(cid:104) f sin q cos q (cid:105)| can have an important effecton (5.46) only when the term (cid:107) f sin q (cid:107) is small. Moreover, using | m x | ∼ we see that this is equivalent to sin q x = v x / | m x | ∼ (cid:104) v (cid:105) being small. Since (cid:104) v (cid:105) (cid:38) Im z , for | z | ≤ ∼ , the imaginary part of z will also be small in therelevant regime.Writing the imaginary part of the QVE in terms of sin q = v/ | m | , we get sin q = | m | Im z + F sin q . (5.49)Since we are interested in a regime where Im z is small, this implies, recalling F f = f , that sin q will then almost lie in the span of f . To make this explicit,we decompose sin q = α f + (Im z ) t , with α = (cid:104) f, sin q (cid:105) , (5.50)for some t ∈ B satisfying (cid:104) f, t (cid:105) = 0 . Let Q (0) denote the orthogonal projection Q (0) w := w − (cid:104) f, w (cid:105) f . Solving for t in (5.49) yields: t = (Im z ) − Q (0) sin q = (1 − F ) − Q (0) | m | . (5.51)Proposition 5.3 implies Gap( F ) ∼ . Therefore we have (cid:107) Q (0) (1 − F ) − Q (0) (cid:107) L → L (cid:46) Gap( F ) − ∼ . In fact, since f x ∼ , a formula analogous to (5.43) applies, and thus we find (cid:107) Q (0) (1 − F ) − Q (0) (cid:107) (cid:46) . Applying this in (5.51) yields (cid:107) t (cid:107) (cid:46) , and therefore sin q = αf + O B (Im z ) . (5.52)Moreover, since we will later use the smallness of (cid:104) v (cid:105) ∼ sin q x ∼ α , we mayexpand cos q = (sign cos q ) ( 1 − sin q ) / = sign Re m + O B ( α ) . (5.53)Combining this with (5.52) yields (cid:10) f sin q cos q (cid:11) = (cid:68) f (cid:0) αf + O B (Im z ) (cid:1)(cid:0) sign Re m + O B ( α ) (cid:1)(cid:69) = σ α + O (cid:0) (cid:104) v (cid:105) + Im z (cid:1) , (5.54)where we have again used α ∼ (cid:104) v (cid:105) , and used the definition, σ = (cid:104) f sign(Re m ) (cid:105) ,from the statement of the lemma.For the term − (cid:107) F (cid:107) L → L in the denominator of the r.h.s. of the mainestimate (5.46) we make use of the explicit formula (5.20) for the spectral radiusof F , − (cid:107) F (cid:107) L → L = Im zα (cid:104) f | m |(cid:105) . (5.55) 50y Proposition 5.3 we have f x ∼ , | m x | ∼ and Gap( F ) ∼ . Using thisknowledge in combination with (5.54), (5.55) and α ∼ (cid:104) v (cid:105) we estimate the r.h.s.of (5.46) further: (cid:107) B − (cid:107) L → L (cid:46) (cid:104) v (cid:105)(cid:104) v (cid:105) + (cid:104) f | m |(cid:105) Im z + (cid:12)(cid:12) σ (cid:104) v (cid:105) + O (cid:0) (cid:104) v (cid:105) + (cid:104) v (cid:105) Im z (cid:1)(cid:12)(cid:12) . (5.56)Let us now see how from this and (5.43) the claim (5.40b) follows. Clearly,it suffices to consider only the case where (cid:104) v (cid:105) ≤ ε for some ε ∼ . If (cid:104) v (cid:105) ≥ | σ | ,then the (cid:104) v (cid:105) -term in the denominator is alone suffices for the final result. Wemay therefore assume that (cid:104) v (cid:105) ≤ | σ | . We are also done if Im z ≥ | σ |(cid:104) v (cid:105) sincethen we may use the second summand on the r.h.s. of (5.56) to get the | σ |(cid:104) v (cid:105) -term we need for (5.40b). In particular, we can assume that the error term in(5.56) is O (cid:0) | σ |(cid:104) v (cid:105) (cid:1) . The bound (5.40b) thus follows by choosing ε ∼ smallenough.We will now show that the perturbed QVE (5.35) is stable as long as a prioribound on m and B − is available. Lemma 5.11 (Stability when m and B − bounded) . Assume A1 . Suppose g, d ∈ B , with inf x | g x | > , satisfy the perturbed QVE (5.35) at some point z ∈ H . Assume (cid:107) m ( z ) (cid:107) ≤ Φ , and (cid:107) B ( z ) − (cid:107) ≤ Ψ , (5.57) for some constants Φ , Ψ ≥ . There exists a linear operator J ( z ) acting on B ,and depending only on S and a in addition to z , with (cid:107) J ( z ) (cid:107) ≤ , such that if (cid:107) g − m ( z ) (cid:107) ≤
12 max { , (cid:107) S (cid:107)} ΦΨ , (5.58) then the correction g − m ( z ) satisfies (cid:107) g − m ( z ) (cid:107) ≤ (cid:107) d (cid:107) (5.59a) |(cid:104) w, g − m ( z ) (cid:105)| ≤
12 max { , (cid:107) S (cid:107)} Ψ Φ (cid:107) w (cid:107) (cid:107) d (cid:107) (5.59b) + Ψ Φ |(cid:104) J ( z ) w, d (cid:105)| , for any w ∈ B . Proof.
Expressing (5.37) in terms of h = g − m = | m | u , and re-arranging weobtain h = | m | B − (cid:2) e − i q h Sh + ( | m | + e − i q h ) d (cid:3) . (5.60)Taking the B -norm of (5.60) yields (cid:107) h (cid:107) ≤ Φ Ψ (cid:107) S (cid:107)(cid:107) h (cid:107) + (Φ Ψ + ΦΨ (cid:107) h (cid:107) ) (cid:107) d (cid:107) . (1 / (cid:107) h (cid:107) and (3 /
2) Φ Ψ (cid:107) d (cid:107) , respectively. Rearranging thus yields (5.59a).In order to prove (5.59b) we apply the linear functional u (cid:55)→ (cid:104) w, u (cid:105) on (5.60),and get |(cid:104) w, h (cid:105)| ≤ (cid:12)(cid:12)(cid:10) w, | m | B − (e i q h Sh ) (cid:11)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) w, | m | B − (e i q hd ) (cid:11)(cid:12)(cid:12) + ΨΦ (cid:12)(cid:12) (cid:104) J w, d (cid:105) (cid:12)(cid:12) , (5.61)where we have identified the operator J := (ΨΦ ) − | m | ( B − ) ∗ ( | m | · ) from thestatement. Clearly, B ∗ is like B except the angle function q is replaced by − q in the definition (5.38). In particular, (cid:107) ( B ∗ ) − (cid:107) ≤ Ψ , and thus (cid:107) J (cid:107) ≤ . Theestimate (5.59b) now follows by bounding the first two term on the right handside of (5.61) separately: (cid:12)(cid:12)(cid:10) w, | m | B − (e i q h Sh ) (cid:11)(cid:12)(cid:12) ≤ (cid:107) w (cid:107) (cid:13)(cid:13) | m | B − (e i q h Sh ) (cid:13)(cid:13) ≤ (cid:107) S (cid:107) Φ Ψ (cid:107) w (cid:107) (cid:107) d (cid:107) (cid:12)(cid:12)(cid:10) w, | m | B − (e i q hd ) (cid:11)(cid:12)(cid:12) ≤ (cid:107) w (cid:107) (cid:13)(cid:13) | m | B − (e i q hd ) (cid:13)(cid:13) ≤ Ψ (cid:107) w (cid:107) (cid:107) d (cid:107) . (5.62)For the rightmost estimates we have used (5.59a) to get (cid:107) h Sh (cid:107) ≤ (cid:107) S (cid:107)(cid:107) h (cid:107) ≤ (cid:107) S (cid:107) Φ Ψ (cid:107) d (cid:107) , and (cid:107) hd (cid:107) ≤ Ψ (cid:107) d (cid:107) , respectively. Now plugging (5.62) into(5.61) and recalling Φ , Ψ ≥ yields (5.59b).52 hapter 6 Uniform bounds
Our main results, such as Theorem 2.6 rely on the assumption that the solution m of the QVE is uniformly bounded. In other words, we assume that there isan upper bound Φ < ∞ , such that ||| m ||| R ≤ Φ , (6.1)and our results deteriorate as Φ becomes larger. In this chapter we introducetwo sufficient quantitative conditions, B1 and B2 on a and S that make itpossible to to construct a constant Φ < ∞ in (6.1) that depend on S and a only through a few model parameters. These extra conditions will always beassumed in conjunction with the properties A1 and A2 .To this end, we introduce a strictly increasing auxiliary function Γ : [0 , ∞ ) → [0 , ∞ ) , determined by a and S : Γ( τ ) := inf x ∈ X (cid:115)(cid:90) X (cid:16) τ + | a y − a x | + (cid:107) S y − S x (cid:107) (cid:17) − π (d y ) . (6.2)We also define the upper limit on the range of Γ , Γ( ∞ ) := lim τ →∞ Γ( τ ) . (6.3)As a strictly increasing function Γ has an inverse Γ − defined on (0 , Γ( ∞ )) .This inverse satisfies Γ − ( λ ) > λ , for < λ < ∞ , and we extend it to ( 0 , ∞ ) by setting Γ − ( λ ) := ∞ , when λ ≥ Γ( ∞ ) .The function Γ( τ ) will be used to convert L bounds on m ( z ) into uniformbounds. We will consider the cases a = 0 and a (cid:54) = 0 separately.When a = 0 Lemma 4.5 implies (cid:107) m ( z ) (cid:107) ≤ | z | − , and hence we only needto obtain an additional L -estimate for m ( z ) around z = 0 . To this end, weintroduce the following condition: B1 Quantitative block fully indecomposability:
There exist two constants ϕ > , K ∈ N , a fully indecomposable matrix Z = ( Z ij ) Ki,j =1 , with Z ij ∈ { , } ,53nd a measurable partition I := { I j } Kj =1 of X , such that for every ≤ i, j ≤ K the following holds: π ( I j ) = 1 K , and S xy ≥ ϕZ ij , whenever ( x, y ) ∈ I i × I j . (6.4)Here the constants ϕ, K are the model parameters associated to B1 . The prop-erty B1 amounts to a quantitative way of requiring S to be a block fully inde-composable operator (cf. Definition 2.9). We also remark that B1 implies A3 by the part (iii) of Proposition 6.9 and the estimate (6.29) below.Our main result concerning the uniform boundedness in the case a = 0 isthe following: Theorem 6.1 (Quantitative uniform bounds when a = 0 ) . Suppose a = 0 , andassume S satisfies A1 and A2 . Then the following uniform bounds hold: (i) Neighborhood of zero:
If additionally B1 holds, then there are con-stants δ > and Φ < ∞ , both depending only on S only through theparameters ϕ, K , s.t., (cid:107) m ( z ) (cid:107) ≤ Φ , for | z | ≤ δ . (6.5)(ii) Away from zero: (cid:107) m ( z ) (cid:107) ≤ | z | − (cid:16) | z | (cid:17) , for | z | > (cid:112) Γ( ∞ ) . (6.6) In particular, if S satisfies B1 and Γ( ∞ ) > δ − , then ||| m ||| R ≤ max (cid:26) Φ , δ − (cid:16) δ (cid:17)(cid:27) , (6.7) where δ and Φ are from (6.5) . The condition in (i) for the bound around z = 0 is optimal for block operatorsby Theorem A.4 below. In Section 11.3 we have collected simple examples thatdemonstrate how the solution can become unbounded around z = 0 when thecondition B1 does not hold. In order to demonstrate the role of Γ in the part(ii) of the theorem we demonstrate in Section 11.2 that some components of thesolution of the QVE may blow up even when A1-3 hold uniformly.
Remark / -Hölder continuous rows when a = 0 ) . Consider thesetup ( X , π ) = ([0 , , d x ) with a = 0 . Assume S satisfies A1-2 , and that itsrows x (cid:55)→ S x ∈ L are piecewise / -Hölder continuous, such that (2.25) holdsfor some finite partition { I k } of [0 , with min k | I k | > . Since the function τ (cid:55)→ | τ | − is not integrable around τ = 0 the range of Γ is unbounded, i.e.,54 ( ∞ ) = ∞ . Therefore applying the part (ii) of Theorem 6.1 we obtain for any δ > the uniform bound (cid:107) m (cid:107) R \ [ − δ,δ ] ≤ δ exp(2 C δ − ) C (cid:112) min k | I k | , where the constant C is from (2.25).The next remark gives a simple example of a block fully indecomposable S . Remark a = 0 ) . The part (i) of Theorem 6.1 impliesthat for any S with a positive diagonal the solution of the QVE is boundedaround z = 0 , e.g., if ( X , π ) = ([0 , , d x ) , and there are constants ε, λ > suchthat S xy ≥ ε { | x − y | ≤ λ } , (6.8)then m ( z ) is bounded on a neighborhood of z = 0 , because S satisfies B1 , with K and ϕ depending only on ε and λ .Now we consider the uniform boundedness in the case a (cid:54) = 0 . In this casethe structural L -estimate from Lemma 4.5 covers only the regime | z | > (cid:107) a (cid:107) . Inorder to get L -bounds also in the remaining regime | z | ≤ (cid:107) a (cid:107) , we introduce aweaker version of the assumption (2.4) used in [AEK16b]: B2 Strong diagonal:
There is a constant ψ > , such that (cid:104) w, Sw (cid:105) ≥ ψ (cid:104) w (cid:105) , ∀ w ∈ B , s.t. w x ≥ . (6.9)Here ψ is considered a model parameter. Since (2.24) implies B2 for some ψ > , the property B2 constitutes a quantitative version of (2.24).The following result is a quantitative version of the part (ii) of Theorem 2.10. Theorem 6.4 (Quantitative uniform bound for general a ) . Assume
A1-3 and B2 . Then there exists a constant Ω ∗ ≥ , depending only on the model param-eters (cid:107) S (cid:107) L → B , ρ, L, ψ , such that if Γ( ∞ ) > Ω ∗ , (6.10) then ||| m ||| R ≤ Γ − (Ω ∗ )Ω / ∗ . (6.11)The threshold Ω ∗ is determined explicitly in (6.17) below. The followingremark provides a simple example in which this theorem is applicable. Remark / -Hölder regularity) . Consider the QVEin the setup ( X , π ) = ([0 , , d x ) . Assume A1-2 . If the map x (cid:55)→ ( a x , S x ) :[0 , → R × L is piecewise / -Hölder continuous in the sense of (2.25), thensimilarly as in Remark 6.2 we see that Γ( ∞ ) = ∞ . If S also has a positivediagonal (6.8), then A3 and B2 hold with L , ρ , and ψ depending only on ε and λ . Hence an application of Theorem 6.4 yields a bound ||| m ||| R ≤ Φ , where Φ depends only on the constants C and min k | I k | from (2.25) and the constants λ and ε from (6.8), in addition to the model parameters (cid:107) S (cid:107) L → B , (cid:107) a (cid:107) from A2 .55 .1 Uniform bounds from L -estimates The next result shows that for a fixed x the corresponding component m x ofan L -solution m of the QVE may diverge only if the pair ( a x , S x ) ∈ R × L is sufficiently far away from most of the other pairs ( a y , S y ) , y (cid:54) = x . In orderto state this result we introduce the refined versions of the auxiliary function(6.2), Γ Λ ,x ( τ ) := (cid:115)(cid:90) X (cid:16) τ + | a y − a x | + (cid:107) S y − S x (cid:107) Λ (cid:17) − π (d y ) , (6.12)where Λ ∈ (0 , ∞ ) and x ∈ X are considered parameters. We remark that (6.2)is related to this operator by Γ( τ ) := inf x Γ ,x ( τ ) . Proposition 6.6 (Converting L -estimates to uniform bounds) . Assume A1 and A2 . Suppose the solution of the QVE satisfies an L -bound, (cid:107) m ( z ) (cid:107) ≤ Λ , for some Λ < ∞ and z ∈ H . Then | m x ( z ) | ≤ ( Γ Λ ,x ) − (Λ) , x ∈ X , (6.13) with the convention that the right hand side if ∞ if Λ is out of the range of Γ Λ ,x .In particular, if a = 0 or Λ ≥ , then the simplified estimate holds: (cid:107) m ( z ) (cid:107) ≤ Γ − (Λ )Λ . (6.14) Proof.
Since m solves the QVE we have (cid:12)(cid:12)(cid:12)(cid:12) m y (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) m x − m x + 1 m y (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) m x + a x − a y + (cid:104) S x − S y , m (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) m x (cid:12)(cid:12)(cid:12)(cid:12) + | a y − a x | + (cid:107) S y − S x (cid:107) (cid:107) m (cid:107) , for any x, y ∈ X . Using (cid:107) m (cid:107) ≤ Λ , we obtain Λ ≥ (cid:90) X | m y | π (d y ) ≥ (cid:90) X (cid:18) | m x | + | a y − a x | + (cid:107) S y − S x (cid:107) Λ (cid:19) − π (d y )= Γ Λ ,x ( | m x | ) . (6.15)As Γ Λ ,x ( τ ) is strictly increasing in τ we see from the definition (6.12) that thisis equivalent to (6.13).If a = 0 or Λ ≥ , then we can take the factor Λ − outside from last integralon the first line of (6.15). This yields the estimate Λ − Γ ,x (Λ | m x | ) ≤ Γ Λ ,x ( | m x | ) ≤ Λ . Multiplying by Λ and taking the infimum over x as a parameter of Γ Λ ,x , the leftmost expression reduces to Γ(Λ | m x | ) ≤ Λ . This is equivalent to (6.14).56 roof of the part (ii) and (6.7) of Theorem 6.1. Since a = 0 the struc-tural L -bound (2.9) reads (cid:107) m ( z ) (cid:107) ≤ / | z | . If Γ( ∞ ) > (2 / | z | ) then we mayuse the estimate (6.14) of Proposition 6.6 to convert this L -estimate into anuniform bound, and we obtain (6.6). The bound (6.7) follows by combining thisestimate with the part (i) of the theorem.In order to prove Theorem 6.4 we need an L -bound also when | z | ≤ (cid:107) a (cid:107) .For this purpose we introduce the following estimate that relies on the property B2 . Lemma 6.7 (Quantitative L -bound) . If A1-3 and B2 hold, then sup z ∈ H (cid:107) m ( z ) (cid:107) ≤ (cid:107) S (cid:107) L − (cid:107) S (cid:107) L → B ρ (cid:18) ψ − / (cid:107) S (cid:107) L → B + 2 (cid:107) a (cid:107) + (cid:112) (cid:107) S (cid:107) (cid:19) . (6.16) Proof of Theorem 6.4.
Using Lemma 6.7 we obtain an L -bound (6.16). Wedefine the threshold, Ω ∗ := max (cid:110) , RHS(6.16) (cid:111) (6.17)Applying the simplified estimate (6.14) of Proposition 6.6 yields (6.11). Proof of Lemma 6.7.
Let κ > be a parameter to be fixed later. We willconsider the two regimes | z | ≤ (cid:107) a (cid:107) + κ and | z | ≥ (cid:107) a (cid:107) + κ , separately. Using thestructural L -estimate from Lemma 4.5, we see that (cid:107) m ( z ) (cid:107) ≤ κ , | z | ≥ (cid:107) a (cid:107) + κ . (6.18)Let us now consider the regime | z | ≤ (cid:107) a (cid:107) + κ . Similarly as in (4.32) weestimate the L -norm of | m | S | m | by the spectral norm of the operator F = F ( z ) , inf x ( S | m | ) x (cid:107) m (cid:107) ≤ (cid:107)| m | S | m |(cid:107) ≤ (cid:107) F e (cid:107) ≤ (cid:107) F (cid:107) L → L , (6.19)where e ∈ B with e x = 1 for every x . From (5.20) we know that (cid:107) F (cid:107) L → L ≤ .Let us write S xy = (cid:104) S x (cid:105) P xy , so that P xy π (d y ) , is a probability measure forevery fixed x . By using (6.19) and Jensen’s inequality we get (cid:107) m (cid:107) ≤ sup x (cid:104) S x (cid:105) (cid:104) P x , | m |(cid:105) ≤ sup x (cid:104) S x (cid:105) (cid:68) P x , | m | (cid:69) ≤ sup x (cid:104) S x (cid:105) (cid:68) S x , | m | (cid:69) ≤ sup x (cid:107) S x (cid:107) (cid:104) S x (cid:105) (cid:13)(cid:13)(cid:13) m (cid:13)(cid:13)(cid:13) . (6.20)By writing the last term in terms of the QVE, and using (5.7) to estimate (cid:104) S x (cid:105) ≥ (cid:107) S (cid:107) − L +1 ρ , we obtain (cid:107) m (cid:107) ≤ (cid:107) S (cid:107) L − (cid:107) S (cid:107) L → B ρ (cid:18) | z | + (cid:107) a (cid:107) + (cid:107) Sm (cid:107) (cid:19) . (6.21) 57he last term inside the parenthesis can be bounded using the L -norm of m , (cid:107) Sm (cid:107) = (cid:104) m, S m (cid:105) ≤ sup x,y ( S ) xy |(cid:104) m (cid:105)| ≤ (cid:107) S (cid:107) → B (cid:104)| m |(cid:105) . (6.22)Here we have used (5.18) for the last inequality. In order to bound the L -norm,we use the property B2 to obtain (cid:104)| m |(cid:105) ≤ (cid:104)| m | , S | m |(cid:105) ψ ≤ (cid:107) F (cid:107) L → L ψ ≤ ψ − . (6.23)Here we have again expressed the norm of m in terms of F and used (cid:107) F (cid:107) L → L ≤ . Using (6.23) in (6.22), and plugging the resulting bound into (6.21), we seethat (cid:107) m ( z ) (cid:107) ≤ (cid:107) S (cid:107) L − (cid:107) S (cid:107) L → B ρ (cid:18) ψ − / (cid:107) S (cid:107) L → B + 2 (cid:107) a (cid:107) + κ (cid:19) , (6.24)for every | z | ≤ (cid:107) a (cid:107) + κ . Choosing κ := (cid:112) (cid:107) S (cid:107) and using (5.7) we see that(6.18) and (6.24) yield (6.16). z = 0 when a = 0 In this section we prove the part (i) of Theorem 6.1. It is clear from Lemma 5.4and (2.9) that Re z = 0 is a special point for the QVE when a = 0 . From(4.27) we read that in this case the real and imaginary parts of the solution m of the QVE are odd and even functions of Re z with fixed Im z , respectively. Inparticular, Re m (i η ) = 0 for η > , and therefore the QVE becomes an equationfor v = Im m alone, v (i η ) = η + Sv (i η ) , ∀ η > . (6.25)It is therefore not surprising that there is a connection between the well posed-ness of the QVE at z = 0 and the question of whether S is scalable . We call S scalable if there exists a positive measurable function h on X , such that h x ( Sh ) x = 1 , ∀ x ∈ X . (6.26)In other words, there exists a positive diagonal operator H such that HSH isdoubly stochastic. In the discrete setup this scalability has been widely studied,see for example Theorem A.5 borrowed from [SK67]. The continuous setuphas been considered in [BLN94]. Here we will show that (cid:107) h (cid:107) (cid:46) , where thecomparison relation is defined w.r.t. the model parameters ( (cid:107) S (cid:107) L → B , ϕ, K ) ,with ϕ and K given in B1 . In order to prove the assertion (i) of Theorem 6.1we use the fact that the solution of the QVE at Re z = 0 is a minimizer of afunctional on positive integrable functions L , where L p + := (cid:8) w ∈ L p : w x > , for π -a.e. x ∈ X (cid:9) , p ∈ [1 , ∞ ] . (6.27) 58 emma 6.8 (Characterization as minimizer) . Suppose S satisfies A1-2 and η > . Then the imaginary part v (i η ) = Im m (i η ) of the solution of the QVEis π -almost everywhere on X equal to the unique minimizer of the functional J η : L → R , J η ( w ) := (cid:104) w, Sw (cid:105) − (cid:104) log w (cid:105) + 2 η (cid:104) w (cid:105) , (6.28) i.e., J η ( v (i η )) = inf w ∈ L J η ( w ) . The characterization of the solution of the continuous scalability problem asa minimizer has been used with η = 0 in [BLN94].We will use the following well known properties of FID matrices. Proposition 6.9 (Properties of FID matrices [BR97]) . Let T = ( T ij ) Ki,j =1 be asymmetric FID matrix. Then the following holds: (i) If P is a permutation matrix, then PT and TP are FID; (ii) There exists a permutation matrix P such that ( TP ) ii > for every i =1 , . . . , K ; (iii) ( T K − ) ij > , for every ≤ i, j ≤ K . The first two properties are trivial. The property (iii) is equivalent to Theo-rem 2.2.1 in [BR97]. For more information on FID matrices and their relation-ship to some other classes of matrices see Appendix A.3.
Proof of the part (i) of Theorem 6.1.
Since Z is a K -dimensional FIDmatrix with { , } -entries it follows from the part (iii) of Proposition 6.9 that min i,j ( Z K − ) ij ≥ . This implies that S is uniformly primitive, ( S K − ) xy ≥ ϕ K − K (cid:88) i,j =1 ( Z K − ) ij { x ∈ I i , y ∈ I j } . (6.29)Showing the uniform bound (6.5) on m is somewhat involved and hence wesplit the proof into two parts. First we consider the case Re z = 0 and show thatthe solution of the QVE, m (i η ) = i v (i η ) , is uniformly bounded. Afterwards weuse a perturbative argument, which allows us to extend the uniform bound on m to a neighborhood of the imaginary axis.Because of the trivial bound v (i η ) ≤ (cid:107) m (i η ) (cid:107) ≤ η − , we restrict ourselvesto the case η ≤ . Step 1 (Uniform bound at Re z = 0 ): Here we will prove sup η> (cid:107) v (i η ) (cid:107) (cid:46) , (6.30) 59here by Convention 5.2 the constants ϕ and K are considered as additionalmodel parameters. As the first step we show that it suffices to bound the averageof v only, since (cid:107) v (i η ) (cid:107) (cid:46) (cid:104) v (i η ) (cid:105) , ∀ η ∈ (0 , . (6.31)In order to obtain (6.31) we recall (5.8) and use Jensen’s inequality similarly asin (6.20), to get (cid:82) X S xy v y π (d y ) (cid:46) (cid:90) X S xy v y π (d y ) . This is used for v = v (i η ) together with the QVE on the imaginary axis (cf.(6.25)) in the chain of inequalities, v = 1 η + Sv ≤ Sv (cid:46) S (cid:16) v (cid:17) = S ( η + Sv ) ≤ η + S v (cid:46) η + (cid:104) v (cid:105) . (6.32)In the last inequality we used the uniform upper bound (5.18) on the integralkernel of S . This establishes (6.31).In order to bound (cid:104) v (cid:105) we argue as follows: First we note that (cid:104) v (cid:105) ≤ K max i =1 (cid:104) v (cid:105) i . (6.33)Here we defined local averages, (cid:104) w (cid:105) i := K (cid:90) I i w x π (d x ) , ∀ i = 1 , . . . , K , (6.34)for any w ∈ L , noting π ( I i ) = K − . Let us also introduce a discretized version (cid:101) J : (0 , ∞ ) K → R of the functional J η by (cid:101) J ( w ) := ϕK K (cid:88) i,j =1 w i Z ij w j − K (cid:88) i =1 log w i , w = ( w i ) Ki =1 ∈ (0 , ∞ ) K , (6.35)where the matrix Z and the model parameter ϕ > are from B1 . The dis-cretized functional is smaller than J η , in the following sense: (cid:101) J ( (cid:104) w (cid:105) , . . . , (cid:104) w (cid:105) K ) (cid:46) J η ( w ) , ∀ w ∈ B , w > . (6.36)To see this we use B1 to estimate S xy ≥ ϕZ ij , ( x, y ) ∈ I i × I j , for the quadraticterm in the definition (6.28) of J η . Moreover, we use Jensen’s inequality to movethe local average inside the logarithm. In other words, (6.36) follows, since J η ( w ) ≥ ϕ K (cid:88) i,j =1 π ( I i ) (cid:104) w (cid:105) i Z ij π ( I j ) (cid:104) w (cid:105) j − K (cid:88) i =1 π ( I i ) (cid:104) log w (cid:105) i ≥ K (cid:40) ϕK K (cid:88) i,j =1 (cid:104) w (cid:105) i Z ij (cid:104) w (cid:105) j − K (cid:88) i =1 log (cid:104) w (cid:105) i (cid:41) = 1 K (cid:101) J ( (cid:104) w (cid:105) , . . . , (cid:104) w (cid:105) K ) , (6.37) 60or an arbitrary w ∈ L . Since K ∈ N is considered a model parameter in thestatement (ii) of Theorem 6.1 the estimate (6.36) follows.Now, by Lemma 6.8 the solution v = v (i η ) of the QVE at z = i η is the(unique) minimizer of the functional J η : L → R . In particular, it yields asmaller value of the functional than the constants function, and thus J η ( v ) ≤ J η (1) = 1 + 2 η ≤ . Combining this with (6.36) we see that (cid:101) J ( (cid:104) v (cid:105) , . . . , (cid:104) v (cid:105) K ) ≤ K ∼ . (6.38)Now we apply the following lemma which relies on Z being FID. The lemma isproven in Appendix A.4. Lemma 6.10 (Uniform bound on discrete minimizer) . Assume w := ( w i ) Ki =1 ∈ (0 , ∞ ) K satisfies (cid:101) J ( w ) ≤ Ψ , for some Ψ < ∞ , where (cid:101) J : (0 , ∞ ) K → R is defined in (6.35) . Then there is aconstant Φ < ∞ depending only on (Ψ , ϕ, K ) , such that K max k =1 w k ≤ Φ . (6.39)From (6.38) we see that we can apply Lemma 6.10 to the discretized vector v := ( (cid:104) v (cid:105) , . . . , (cid:104) v (cid:105) K ) , with Ψ := 3 K ∼ , and obtain: K max i =1 (cid:104) v (cid:105) i (cid:46) . Plugging this into (6.33) and the resulting inequality for (cid:104) v (cid:105) into (6.31) yieldsthe chain of bounds, (cid:107) v (cid:107) (cid:46) (cid:104) v (cid:105) ≤ max k (cid:104) v (cid:105) k (cid:46) . This completes the proof of(6.30) Step 2 (Extension to a neighborhood):
It remains to show that thereexists δ ∼ , such that (cid:107) m ( τ + i η ) − m (i η ) (cid:107) (cid:46) | τ | , when | τ | ≤ δ . (6.40)Here Φ := sup η (cid:107) m (i η ) (cid:107) < ∞ is considered a model parameter. In particular, thebound (5.9) on | m (i η ) | = v (i η ) implies v (i η ) ∼ . By (5.40b) of Lemma 5.9 wefind (cid:107) B (i η ) − (cid:107) (cid:46) . The bound (6.40) follows now from Lemma 5.11 by choosing z = i η and d x = τ . Indeed, the lemma states that with the abbreviation h ( τ ) := m (i η + τ ) − i v (i η ) , the following holds true. If (cid:107) h ( τ ) (cid:107) ≤ c for a sufficiently small constant c ∼ ,then actually (cid:107) h ( τ ) (cid:107) ≤ C | τ | for some large constant C depending only on Φ and the other model parameters.The Stieltjes transform representation (2.8) implies that h ( τ ) is a continuousfunction in τ . As h (0) = 0 , by definition, the bound (cid:107) h ( τ ) (cid:107) ≤ C | τ | applies aslong as C | τ | ≤ c remains true. With the choice δ := c /C we finish the proofof (6.5). 61 hapter 7 Regularity of solution
We will now estimate the complex derivative ∂ z m on the upper half plane H .When ||| m ||| R < ∞ these bounds turn out to be uniform in z . This makes itpossible to extend the domain of the map z (cid:55)→ m ( z ) to the closure H = H ∪ R .Additionally, we prove that the solution and its generating density are / -Hölder continuous (Proposition 7.1), and analytic (Corollary 7.6) away fromthe special points τ ∈ supp v where v ( τ ) = 0 . Combining these two resultswe prove Theorem 2.4 at the end of this chapter. Even if the uniform bound, ||| m ||| R < ∞ , is not available we still obtain weaker regularity for the averagedsolution (cid:104) m (cid:105) . The analyticity of the solution of the QVE is not restricted tothe variable z alone. In Proposition 7.5 we show that the QVE perturbed by asmall element d ∈ B still has a unique solution g = g ( z, d ) close to m ( z ) thatdepends analytically on d provided z is not close to a point τ ∈ supp v with v ( τ ) = 0 .At the technical level, the proofs of both the Hölder-continuity and theanalyticity of m boil down to considering small, in fact infinitesimally small,perturbations of the QVE and then applying the estimates from Section 5.2. Proposition 7.1 (Hölder regularity in z and extension to real line) . Assume
A1-3 . For an interval I ⊂ R and a constant ε > , set D := (cid:8) z ∈ H : dist( z, [ − ,
2Σ ] \ I ) ≥ ε (cid:9) . (7.1) Then the following hold: (i)
If there is Λ < ∞ , such that (cid:107) m ( z ) (cid:107) ≤ Λ , Re z ∈ I , (7.2) then the averaged solution of the QVE is uniformly Hölder-continuous, |(cid:104) m ( z ) (cid:105) − (cid:104) m ( z ) (cid:105)| (cid:46) | z − z | / , z , z ∈ D , (7.3) where ε and Λ are considered additional model parameters. If (7.2) is replaced by the uniform bound, ||| m ||| I ≤ Φ < ∞ , then the Höldercontinuity is improved to (cid:107) m ( z ) − m ( z ) (cid:107) (cid:46) | z − z | / , z , z ∈ D , (7.4) where ε and Φ are considered additional model parameters. We remark that if S satisfies B2 (cf. Chapter 6), in addition to A1-3 , thenLemma 6.7 provides an effective upper bound Λ for the L -norm of m ( z ) , with I = R . Similarly, quantitative uniform bounds can be obtained using Theo-rem 6.1 and Theorem 6.4. In a slightly different setup a qualitative version ofthe / -Hölder continuity (7.4) was established in [AEK16b] as Proposition 5.1. Convention 7.2 (Extension to real axis) . When m is uniformly bounded ev-erywhere, i.e., ||| m ||| R < ∞ , then (7.4) guarantees that m can be extended to thereal axis. We will then automatically consider m , and all the related quantitiesas being defined on the extended upper half plane H = H ∪ R .In the proof of the part (ii) of Proposition 7.1 we actually show the followingestimate on the derivative of m ( z ) . Corollary 7.3 (Bound on derivative) . In Proposition 7.1 the inequality (7.4) can be replaced by a stronger bound, (cid:0) | σ |(cid:104) Im m (cid:105) + (cid:104) Im m (cid:105) (cid:1) (cid:107) ∂ z m (cid:107) ≤ C , on D . Here the function σ is from (5.41) and C depends on the model parametersfrom the part (ii) of Proposition 7.1. The proof of Proposition 7.1 also yields a regularity result for the meangenerating measure when a = 0 . Corollary 7.4 (Regularity of mean generating density) . Assume
A1-3 , andsuppose a = 0 . Then the normalized mean generating measure ν (d τ ) := 1 π (cid:104) v (d τ ) (cid:105) , (7.5) has the representation ν (d τ ) = (cid:101) ν ( τ ) d τ + ν δ (d τ ) , (7.6) where ≤ ν ≤ , and the Lebesgue-absolutely continuous part (cid:101) ν ( τ ) is symmetricin τ , and locally Hölder-continuous on R \{ } . More precisely, for every ε > , | (cid:101) ν ( τ ) − (cid:101) ν ( τ ) | (cid:46) | τ − τ | / , ∀ τ , τ ∈ R \ ( − ε, ε ) , (7.7) where ε is an additional model parameter.If additionally, B1 holds then ν = 0 in (7.6) and (7.7) holds for all τ , τ ∈ R with C depending only on the model parameters from A1-2 and B1 . roof. As an intermediate step of the proof of Proposition 7.1 below, we iden-tify (cid:101) ν | I as the uniformly / -Hölder continuous extension of (cid:104) v (cid:105) to any realinterval I such that (7.2) holds.Let us now assume A1-3 , and fix some ε > . By (2.9) we have the uniform L -estimate (cid:107) m ( z ) (cid:107) ≤ ε − , for z ∈ H satisfying | Re z | ≥ ε . In other words,the hypothesis (7.2) of Proposition 7.1 holds with Λ = 2 ε − and I := R \ ( − ε, ε ) ,and thus both (7.6) and (7.7) follow.If B1 is assumed in addition to A1-3 , then the part (i) of Theorem 6.1implies that (cid:107) m ( z ) (cid:107) ≤ Φ when | Re z | ≤ δ , for some Φ , δ ∼ . Combiningthis with the L -estimate (cid:107) m ( z ) (cid:107) ≤ /δ valid for | Re z | ≥ δ , we see thatProposition 7.1 is applicable with I = R and Λ := max { Φ , δ − } . Proof of Proposition 7.1.
The solution m is a holomorphic function from H to B by Theorem 2.1. In particular, if | z | > , then the claims of theproposition follow trivially from (2.8) and (2.7). Thus we will assume | z | ≤ here.Taking the derivative with respect to z on both sides of (2.4) yields (1 − m ( z ) S ) ∂ z m ( z ) = m ( z ) , ∀ z ∈ H . Expressing this in terms of the operator B = B ( z ) from (5.38), and suppressingthe explicit z -dependence, we obtain i2 ∂ z v = ∂ z m = | m | B − | m | . (7.8)Here we have also used the general property ∂ z φ = i2 ∂ z Im φ , valid for all ana-lytic functions φ : K → C , K ⊂ C , to replace m by v = Im m . Case 1 (No uniform bound on m ): Consider z ∈ H satisfying | z | ≤ and Re z ∈ I . Taking the average of (7.8) yields ∂ z (cid:104) v (cid:105) = (cid:104) | m | , B − | m | (cid:105) , where v = Im m by (4.21), and thus (cid:12)(cid:12) ∂ z (cid:104) v (cid:105) (cid:12)(cid:12) ≤ − (cid:107) m (cid:107) (cid:107) B − (cid:107) L → L (cid:107) m (cid:107) (cid:46) (cid:104) v (cid:105) − , Re z ∈ I . (7.9)In the last step we used (5.39) to get (cid:107) B − (cid:107) L → L (cid:46) (cid:104) v (cid:105) − . This is wherethe assumption (7.2) was utilized. The bound (7.9) implies that z (cid:55)→ (cid:104) v ( z ) (cid:105) isuniformly / -Hölder-continuous when Re z ∈ I . Consequently, the probabilitymeasure ν has a Lebesgue-density on I , (cid:101) ν ( τ ) = ν (d τ )d τ = 1 π lim η ↓ (cid:104) v ( τ + i η ) (cid:105) , τ ∈ I , (7.10)and this density inherits the uniform Hölder continuity from (7.9).It remains to extend this regularity from the mean generating measure ν | I to its Stieltjes transform (cid:104) m (cid:105)| D . To this end, let us denote the left and right end64oints of the real interval I by τ − and τ + , respectively. Let us split ν , into twonon-negative measures, ν = ν + ν . Here the first measure is defined by ν (d τ ) := ϕ ( τ ) ν (d τ ) , with the function ϕ : R → [0 , , being a piecewise linear such that, ϕ ( τ ) = 0 for τ ∈ R \ [ τ − + ε/ , τ + − ε/ , ϕ ( τ ) = 1 when τ − + (2 / ε ≤ τ ≤ τ + − (2 / ε , and linearlyinterpolating in between. It follows, that ν has a Lebesgue-density (cid:101) ν and issupported in [ − Σ , Σ ] , since supp v ⊆ [ − Σ , Σ ] by Theorem 2.1. Furthermore, | (cid:101) ν ( τ ) − (cid:101) ν ( τ ) | (cid:46) | τ − τ | / , ∀ τ , τ ∈ R . (7.11)For the measure ν we know that ν ( R ) ≤ ν ( R ) = 1 , and supp ν ⊆ (cid:2) − Σ , τ − + ε (cid:3) ∪ (cid:2) τ + − ε , Σ (cid:3) , where one of the intervals may be empty, i.e., [ τ (cid:48) , τ (cid:48)(cid:48) ] := ∅ , for τ (cid:48) > τ (cid:48)(cid:48) . TheStieltjes transform (cid:104) m ( z ) (cid:105) = (cid:90) R ν (d τ ) τ − z , is a sum of the Stieltjes transforms of ν and ν . The Stieltjes transform of ν isHölder-continuous with Hölder-exponent / since this regularity is preservedunder the Stieltjes transformation. For the convenience of the reader, we statethis simple fact as Lemma A.7 in the appendix. On the other hand, since Re z is away from the support of ν , the Stieltjes transform of ν satisfies (cid:12)(cid:12)(cid:12) ∂ z (cid:90) X ν (d τ ) τ − z (cid:12)(cid:12)(cid:12) ≤ ε (cid:46) , when z ∈ D , and hence (7.3) follows. Case 2 (solution uniformly bounded): Now we make the extra assumption ||| m ||| I ≤ Φ ∼ , I := [ τ − , τ + ] ⊆ R . Taking the B -norm of (7.8) immediatelyyields | ∂ z v x ( z ) | ≤ (cid:107) m ( z ) (cid:107) (cid:107) B ( z ) − (cid:107) (cid:46) (cid:104) v ( z ) (cid:105) − ∼ v x ( z ) − . (7.12)Here we used (5.40b) to estimate the norm of B − , and the part (ii) of Propo-sition 5.3 to argue that v ( z ) ∼ (cid:104) v ( z ) (cid:105) . We see that z (cid:55)→ v x ( z ) is / -Höldercontinuous uniformly in z ∈ I + i (0 , ∞ ) and x ∈ X . Repeating the localizationargument used to extend the regularity of (cid:101) ν = π − (cid:104) v (cid:105) to the correspondingStieltjes transform yields (7.4). Proof of Corollary 7.3.
Using all the terms of (5.40b) for the second boundof (7.12) and using (7.8) to estimate | ∂ z m | ∼ | ∂ z v | yields the derivative boundof the corollary. 65ext we show that the perturbed QVE (2.29) has a unique solution. For thestatements of this result we introduce a shorthand D B ( h, ρ ) := (cid:8) g ∈ B : (cid:107) g − h (cid:107) < ρ (cid:9) , for the open B -ball centred at h with radius ρ > . We also recall that forcomplex Banach spaces X and Y , a map φ : U → Y is called holomorphic onan open set U ⊂ X if for every x ∈ U , every x ∈ X , and every bounded linearfunctional γ ∈ X (cid:48) the map ζ (cid:55)→ (cid:104) γ, φ ( x + ζ x ) (cid:105) : C → C defines a holomorphicfunction in a neighborhood of ζ = 0 . This is equivalent (cf. Section 3.17 of[HP57]) to the existence of a Fréchet-derivative of φ on U , i.e., for every x ∈ U there exists a bounded complex linear operator Dφ ( x ) : X → Y , such that (cid:107) φ ( x + d ) − φ ( x ) − Dφ ( x ) d (cid:107) Y (cid:107) d (cid:107) X → , as (cid:107) d (cid:107) X → . Proposition 7.5 (Analyticity) . Assume
A1-3 , and consider a fixed z ∈ H satisfying | z | ≤ , where Σ := (cid:107) a (cid:107) + 2 (cid:107) S (cid:107) / , such that (cid:107) m ( z ) (cid:107) ≤ Φ , and (cid:107) B ( z ) − (cid:107) ≤ Ψ , (7.13) for some constants Φ < ∞ and Ψ ≥ . Let us define ε := 13 Σ + 9 (cid:107) S (cid:107) ΦΨ , and δ := ε Ψ . (7.14) Then there exists a holomorphic map d (cid:55)→ g ( z, d ) : D B (0 , δ ) → D B ( m ( z ) , ε ) ,where g = g ( z, d ) is the unique solution of the perturbed QVE, − g = z + a + Sg + d , (7.15) in D B ( m ( z ) , ε ) . The Fréchet-derivative Dg ( z, d ) of g ( z, d ) w.r.t. d is uniformlybounded, (cid:107) Dg ( z, d ) (cid:107) ≤ / (cid:107) S (cid:107) . In particular, (cid:107) g ( z, d ) − m ( z ) (cid:107) ≤ (cid:107) S (cid:107) (cid:107) d (cid:107) , ∀ d ∈ D B (0 , δ ) . (7.16)Before proving Proposition 7.5 we consider its applications. First we showthat apart from a set of special points the generating measure v has an analyticdensity on the real line. Corollary 7.6 (Real analyticity of generating density) . Assume
A1-3 , andconsider a fixed τ ∈ R . If additionally, either of the following three sets ofconditions are assumed, (i) (cid:104) v ( τ ) (cid:105) > , and B2 holds; (ii) | τ | > (cid:107) a (cid:107) and (cid:104) v ( τ ) (cid:105) > ; τ = 0 , a = 0 , and B1 holds,then the generating density v is real analytic around τ . Proof.
Since ∂ z m ( z ) = Dg ( z, e , where e x = 1 for all x ∈ X , the result followsimmediately from Proposition 7.5 once we have shown that both (cid:107) m ( τ ) (cid:107) < ∞ and (cid:107) B ( τ ) − (cid:107) < ∞ hold. Actually, it suffices to only prove (cid:107) m ( τ ) (cid:107) < ∞ and (cid:104) v ( τ ) (cid:105) > in all the three cases (i)-(iii). Indeed, with these estimates at hand,the bound (5.40b) of Lemma 5.9 yields (cid:107) B ( τ ) − (cid:107) < ∞ .In the case (i) we first use Lemma 6.7 to obtain sup z ∈ H (cid:107) m ( z ) (cid:107) ≤ Λ , for Λ ∼ . We then plug this L -bound in the lower bound of the part (i) ofLemma 5.4 to get a uniform lower bound inf x | m x ( τ ) | (cid:38) Λ − . Using this in theupper bound of the part (i) of Lemma 5.4 yields (cid:107) m ( τ ) (cid:107) (cid:46) Λ − C (cid:104) v (cid:105) − ∼ .In the case (ii) we first note that | τ | > (cid:107) a (cid:107) implies dist( τ, { a y } ) > , andthus the first inequality of (5.9) yields inf x | m x ( τ ) | > . Plugging this into thesecond inequality of (5.9), and using the assumption (cid:104) v ( τ ) (cid:105) > , we obtain anuniform bound for m ( τ ) .In the case (iii), we use the part (ii) of Theorem 6.1 to get the uniformbound. From the symmetry (4.27) we see that m (0) = i v (0) . Hence (5.9) yields inf x v x (0) > . Feeding this into (5.10) yields (cid:104) v (0) (cid:105) ∼ .Combining the analyticity and the Hölder regularity we prove Theorem 2.4. Proof of Theorem 2.4.
Here we assume ||| m ||| R ≤ Φ , with Φ < ∞ consideredas a model parameter. The assertion (i) follows from (ii) of Proposition 5.3.Using the bound (7.4) of Proposition 7.1, with I = R , we see that m can beextended as a / -Hölder continuous function to the real line. Hence, from (4.21)we read off that the generating measure must have a Lebesgue-density equal to Im m | R . In particular, this density function inherits the Hölder regularity from m | R , i.e., for some C ∼ : | v x ( τ (cid:48) ) − v x ( τ ) | ≤ C | τ (cid:48) − τ | / , ∀ τ, τ (cid:48) ∈ R . (7.17)This proves the part (iii) of the theorem.Since ||| m ||| R ∼ using Lemma 5.4 we see that v x ( z ) ∼ v y ( z ) for z ∈ H .Let τ ∈ R be such that v ( τ ) > . In order to bound the derivatives of v at τ we use (7.17) to estimate inf (cid:8) | ω | : v ( τ + ω ) = 0 , ω ∈ R (cid:9) ≥ C − (cid:104) v ( τ ) (cid:105) =: (cid:37) > . By Corollary 7.6 this implies that v is analytic on the ball of radius (cid:37) centeredat τ . The Cauchy-formula implies that the k -th derivative of v at τ is boundedby k ! (cid:37) − k . This proves the assertion (ii) of the theorem. Proof of Proposition 7.5. As z is fixed, we write m = m ( z ) . We start withgeneral ε and δ , i.e., (7.14) is not assumed. Since | z | < , we see directly from67he QVE that / | m | ≤ | z | + (cid:107) a (cid:107) + (cid:107) S (cid:107)(cid:107) m (cid:107) ≤
3Σ + (cid:107) S (cid:107) Φ . Writing | w/m | ≤ (cid:107) /m (cid:107)(cid:107) w − m (cid:107) , we thus find that (cid:12)(cid:12)(cid:12) wm (cid:12)(cid:12)(cid:12) ≤ , ∀ w ∈ D B ( m, ε ) , provided ε ≤
13Σ + (cid:107) S (cid:107) Φ . (7.18)We will assume below that ε satisfies the above condition.Consider now z and d ∈ D B (0 , δ ) fixed. We will first construct a function λ (cid:55)→ g ( λ ) : [0 , → D B ( m, ε ) , such that g ( λ ) solves (7.15) with λd in place of d , i.e., Z ( λ, g ( λ )) = 0 , where Z ( λ, w ) := w + 1 z + a + Sw + λd . (7.19)Let us define R : D B ( m, ε ) → B by R ( w ) := (1 − w S ) − ( w d ) . (7.20)The function λ (cid:55)→ g ( λ ) is obtained by solving the Banach-space valued ODE ∂ λ g ( λ ) = R ( g ( λ )) , λ ∈ [0 , ,g (0) = m , (7.21)where m = m ( z ) . Indeed, a short calculation shows that if λ (cid:55)→ g ( λ ) solves theODE, then dd λ Z ( λ, g ( λ )) = 0 . As Z (0 , g (0)) = 0 by the definition of m ( z ) , this implies that also Z (1 , g (1)) = 0 ,which is equivalent to g = g (1) solving (7.15).We will now find ε, δ ∼ such that (cid:107) R ( w ) (cid:107) ≤ ε , for w ∈ D B ( m, ε ) and (cid:107) d (cid:107) B ≤ δ . Under this condition the elementary theory of ODEs (cf. Theorem9.1 of [Col12]) yields the unique solution g ( λ ) ∈ D B ( m, ε ) to (7.21). We startby estimating the the norm of the following operator (1 − uwS ) − = ( 1 + | m | B − D ) − | m | B − (cid:16)(cid:16) | m | m (cid:17) ·| m | (cid:17) , (7.22)for arbitrary u, w ∈ D B ( m, ε ) . Here, D := ( | m | /m ) m − ( m − uw ) S . Since m − uw = m ( m − u ) + u ( m − w ) we get (cid:107) D (cid:107) ≤ (cid:107) S (cid:107) ε using (7.18). Thusrequiring (cid:107)| m | B − D (cid:107) ≤ (cid:107) S (cid:107) ε to be less than / , we see that (cid:107) ( 1 + | m | B − D ) − (cid:107) ≤ , provided ε ≤
16 ΦΨ (cid:107) S (cid:107) . (7.23)Using this bound for the first factor on the right hand side of (7.22) yields (cid:107) (1 − uwS ) − h (cid:107) ≤ (cid:107) h/m (cid:107) , ∀ h ∈ B , ∀ u, w ∈ D B ( m, ε ) , (7.24) 68rovided the condition for ε in (7.23) holds. In order to estimate (cid:107) R ( w ) (cid:107) for w ∈ D B ( m, ε ) we choose u = w and h = w d in (7.24), and get (cid:107) R ( w ) (cid:107) ≤ (cid:107) w /m (cid:107)(cid:107) d (cid:107) ≤ Ψ δ , (7.25)where (cid:107) d (cid:107) ≤ δ and (cid:107) w /m (cid:107) = (cid:107) w/m (cid:107) (cid:107) m (cid:107) ≤ were used for the last bound.With the choice (7.14) for δ we see that the rightmost expression in (7.25) isless than ε . Moreover, if ε is chosen according to (7.14), then the conditionsfrom the estimates (7.18) and (7.23) are both satisfied as Ψ ≥ . We concludethat the ODE (7.21) has a unique solution in D B ( m, ε ) if we choose ε and δ tosatisfy (7.14).In order to show that not only the ODE but the perturbed QVE (7.15) ingeneral has a unique solution in D B ( m, ε ) , we establish a more general stabilityresult. To this end, assume that g, g (cid:48) ∈ D B ( m ( z ) , ε ) and d, d (cid:48) ∈ D B (0 , δ ) aresuch that g solves (7.15), while g (cid:48) solves the same equation with d replaced by d (cid:48) . Then by definition, (1 − gg (cid:48) S )( g (cid:48) − g ) = gg (cid:48) ( d (cid:48) − d ) . (7.26)Applying (7.24) to (7.26) and recalling | g/m | , | g (cid:48) /m | ≤ we obtain (cid:107) g (cid:48) − g (cid:107) ≤ Ψ (cid:107) d (cid:48) − d (cid:107) . (7.27)The uniqueness of the solution to (7.15) follows now from (7.26). In particu-lar, this implies that the map d (cid:55)→ g ( z, d ) : D B (0 , δ ) → D B ( m ( z ) , ε ) is uniquelydefined with g ( z, d ) := g (1) , where g (1) is the value of the solution of the ODE(7.21) at λ = 1 .It remains to show that g ( z, d ) is analytic in d . To this end, let h ∈ B bearbitrary, and consider (7.26) with g = g ( z, d ) , g (cid:48) = g ( z, d (cid:48) ) , where d (cid:48) = d + ξ h for some sufficiently small ξ ∈ C . Using the stability bound (7.27) we arguethat the differences g − g (cid:48) vanish in the limit ξ → . Therefore we obtain from(7.26) Dg ( z, d ) h := lim ξ → g ( z, d + ξ h ) − g ( z, d ) ξ = (1 − g ( z, d ) S ) − ( g ( z, d ) h ) , where Dg ( z, d ) : B → B is the Fréchet-derivative of g ( z, d ) w.r.t. d at ( z, d ) .69 hapter 8 Perturbations whengenerating density is small
In this chapter we analyze the stability of the QVE (2.4) in the neighborhoodof parameters z with a small value of the average generating density (cid:104) v ( z ) (cid:105) ,against adding a perturbation d ∈ B to the right hand side. In the special casewhen d is a real constant function, i.e., when m ( z ) is compared to m ( z + ω ) ,and when z ∈ supp (cid:104) v (cid:105) with (cid:104) v ( z ) (cid:105) = 0 , this analysis has been carried out in[AEK16b]. In that special case the upcoming proofs simplify considerably forthe following three reasons. First, an expansion in α (cf. Lemma 8.1) is notneeded. Second, we do not need to show that the expansions are uniform inthe model parameters. Third, the complicated selection process of the roots inSubsection 9.2.2 is avoided as we do not have to consider very small gaps in thesupport of the generating density.We will assume in this and the following chapters that S satisfies A1-3 and that the solution is uniformly bounded everywhere ||| m ||| R ≤ Φ < ∞ . Inparticular, all the comparison relations (Convention 2.3) will depend on:(8.1) ’The model parameters’ := ( ρ, L, (cid:107) a (cid:107) , (cid:107) S (cid:107) L → B , Φ) . Due to the uniform boundedness, m and all the related quantities are extendedto H (cf. Proposition 7.1). Furthermore, these standing assumptions also implythat Proposition 5.3 is effective, i.e., | m x ( z ) | , f x ( z ) , Gap( F ( z )) ∼ , and v x ( z ) ∼ (cid:104) v ( z ) (cid:105) ∼ α ( z ) , (8.2)for every | z | ≤ and x ∈ X . In particular, the three quantities v, (cid:104) v (cid:105) , α = (cid:104) f, sin q (cid:105) , can be interchanged at will, as long as only their sizes up to constantsdepending on the model parameters matter.The stability of the QVE against perturbations deteriorates when the gen-erating density becomes small. This can be seen from the explosion in theestimate (cid:104) v ( τ ) (cid:105) − (cid:46) (cid:107) B ( τ ) − (cid:107) (cid:46) (cid:104) v ( τ ) (cid:105) − , τ ∈ supp v | R , (8.3) 70cf. (5.40b) and (8.10b) below) for the inverse of the operator B , introducedin (5.38). This norm appears in the estimates (5.59) relating the norm of therescaled difference, u = g − m | m | , (8.4)of the two solutions g and m of the perturbed and the unperturbed QVE, − g = z + a + Sg + d and − m = z + a + Sm , respectively, to the size of the perturbation d .The unboundedness of B − in (8.3), as (cid:104) v (cid:105) → , is caused by the vanishingof B in a one-dimensional subspace of L corresponding to the eigendirection ofthe smallest eigenvalue of B . Therefore, in order to extend our analysis to theregime (cid:104) v (cid:105) ≈ we decompose the perturbation (8.4) into two parts: u = Θ b + r . (8.5)Here, Θ is a scalar, and b is the eigenfunction corresponding to the smallesteigenvalue of B . The remaining part r ∈ B lies inside a subspace where B − isbounded due to the spectral gap of F (cf. Figure 8.1). As B is not symmetric, r and b are not orthogonal w.r.t. the standard inner product (2.5) on L . Themain result of this chapter is Proposition 8.2 which shows that for sufficientlysmall (cid:104) v (cid:105) ≤ ε ∗ , the b -component Θ of u satisfies a cubic equation, and we identifyits coefficients up to the leading order in the small parameters (cid:104) v (cid:105) and d . Wewill use the symbol ε ∗ ∼ as the upper threshold for (cid:104) v (cid:105) and its value will bereduced along the proofs. B In this section we collect necessary information about the operator B : B → B defined in (5.38). Recall, that the spectral projector P λ corresponding to anisolated eigenvalue λ of a compact operator T acting on a Banach space X isobtained (cf. Theorem 6.17 in Chapter 3 of [Kat12]) by integrating the resolventof T around a loop Γ encircling only the eigenvalue λ : P λ := − π i (cid:73) Γ ( T − ζ ) − d ζ . (8.6) Lemma 8.1 (Expansion of B in bad direction) . There exists ε ∗ ∼ such that,uniformly in z ∈ H with | z | ≤ , the following holds true: If α = α ( z ) = (cid:68) f ( z ) , Im m ( z ) | m ( z ) | (cid:69) ≤ ε ∗ , hen the operator B = B ( z ) has a unique single eigenvalue β = β ( z ) of small-est modulus, so that | β (cid:48) | − | β | (cid:38) , ∀ β (cid:48) ∈ Spec( B ) \{ β } . The correspondingeigenfunction b = b ( z ) , satisfying Bb = β b , has the properties (cid:104) f, b (cid:105) = 1 , and | b x | ∼ , ∀ x ∈ X . (8.7) The spectral projector P = P ( z ) : B → Span { b ( z ) } corresponding to β , is givenby (8.8) P w = (cid:104) b, w (cid:105)(cid:104) b (cid:105) b . Denoting, Q := 1 − P , we have (cid:107) B − (cid:107) (cid:46) α − , but (cid:107) B − Q (cid:107) + (cid:107) ( B − Q ) ∗ (cid:107) (cid:46) , (8.9) where ( B − Q ) ∗ is the L -adjoint of B − Q .Furthermore, the following expansions in η = Im z and α hold true: B = 1 − F − pf α − f α + O B → B ( α + η ) , (8.10a) β = (cid:104) f | m |(cid:105) ηα − i2 σ α + 2 ( ψ − σ ) α + O ( α + η ) , (8.10b) b = f + i 2 (1 − F ) − Q (0) ( pf ) α + O B ( α + η ) . (8.10c) If z ∈ R , then the ratio η/α is defined through its limit η ↓ . The real valuedauxiliary functions σ = σ ( z ) and ψ = ψ ( z ) ≥ in (8.10) , are defined by σ := (cid:104) pf (cid:105) and ψ := D ( pf ) , (8.11) where the sign function p = p ( z ) , and the positive quadratic form D = D ( · ; z ) ,are given by p := sign Re m (8.12) and D ( w ) := (cid:68) Q (0) w, (cid:104) (1 + (cid:107) F (cid:107) L → L )(1 − F ) − − (cid:105) Q (0) w (cid:69) ≥ Gap( F )2 (cid:107) Q (0) w (cid:107) , (8.13) respectively. The orthogonal projector Q (0) = Q (0) ( z ) := 1 − f ( z ) (cid:104) f ( z ) , · (cid:105) is theleading order term of Q , i.e., Q = Q (0) + O L → L ( α ) . Furthermore, Gap( F ) ∼ .Finally, λ ( z ) = (cid:107) F ( z ) (cid:107) L → L , β ( z ) , σ ( z ) , ψ ( z ) , as well as the vectors f ( z ) , b ( z ) ∈ B , are all uniformly / -Hölder continuous functions of z on connected compo-nents of the domain (cid:8) z ∈ H : α ( z ) ≤ ε ∗ , | z | ≤ (cid:9) , where Σ ∼ is from (2.7) . The function p stays constant on these connectedcomponents. P is not an orthogonal projection (unless b = b ), it follows from(8.7) and (8.8) that (cid:107) P (cid:107) , (cid:107) P ∗ (cid:107) (cid:46) . (8.14)Here P ∗ = b (cid:104) b, · (cid:105) / (cid:104) b (cid:105) is the Hilbert space adjoint of P . Proof.
Recall that sin q = (Im m ) / | m | (cf. (5.38)), and B = e − i2 q − F = (1 − F ) + D , (8.15)where D is the multiplication operator D = − i2 cos q sin q − q . (8.16)From the definition of α = (cid:104) f Im m/ | m |(cid:105) , and f, | m | ∼ , we see that | sin q | ∼ α ,and thus (cid:107) D (cid:107) L → L + (cid:107) D (cid:107) ≤ C α , (8.17)for some C ∼ . The formula (8.10a) for B follows by expanding D in α and η using the representations (5.50) and (5.53) of sin q and cos q , respectively. Inparticular, from (5.51) we know that (cid:107) t (cid:107) (cid:46) , and thus sin q = αf + O B ( η ) .Let us first consider the operators acting on the space L . By Proposition 5.3the operator − F has an isolated single eigenvalue of smallest modulus equalto − (cid:107) F (cid:107) L → L = ηα (cid:104)| m | f (cid:105) , (8.18)and the L -spectrum of − F lies inside the set L := (cid:8) − (cid:107) F (cid:107) L → L (cid:9) ∪ (cid:2) − (cid:107) F (cid:107) L → L + Gap( F ) , (cid:3) . (8.19)Here the upper spectral gap of F satisfies Gap( F ) ∼ by (iv) of Proposition 5.3.The properties of β and b , etc., are deduced from the resolvent of B by usingthe analytic perturbation theory (cf. Chapter 7 of [Kat12]). To this end denote R ( ζ ) := (1 − F − ζ ) − , so that ( B − ζ ) − = (1 + R ( ζ ) D ) − R ( ζ ) . We will now bound R ( ζ ) = − ( (cid:98) F ( | m | ) − (1 − ζ )) − as an operator on B , usingthe property (5.28) of the resolvent of the F -like operators (cid:98) F (cf. (5.23)) (cid:107) R ( ζ ) (cid:107) ≤ (cid:107) R ( ζ ) (cid:107) L → L | ζ − | . (8.20)Thus there exists a constant δ ∼ , (cid:107) R ( ζ ) (cid:107) (cid:46) , dist( ζ , L ) ≥ δ . L contains both the L -spectrum of − F ,and the point ζ = 1 . Thus (8.20) shows that L contains also the B -spectrumof − F . By requiring ε ∗ to be sufficiently small it follows from (8.17) that (cid:107) (1 + R ( ζ ) D ) − (cid:107) (cid:46) provided ζ is at least a distance δ away from L , and thus (cid:107) ( B − ζ ) − (cid:107) (cid:46) , dist( ζ , L ) ≥ δ . (8.21)By (iv) in Proposition 5.3 we see that Gap( F ) (cid:38) . By taking ε ∗ sufficiently small the perturbation (cid:107) D (cid:107) becomes so small that we maytake δ ≤ Gap( F ) / . It then follows that the eigenvalue β is sepa-rated from the rest of the B -spectrum of B by a gap of size δ ∼ .Figure 8.1: The spectrum of − F lies inside theunion of an interval with one isolated point. The per-turbation B of − F has spectrum in the indicatedarea.Knowing this sepa-ration, the standardresolvent contour in-tegral representationformulas (cf. (8.6))imply that (cid:107) b (cid:107) (cid:46) and (cid:107) P (cid:107) (cid:46) , (cid:107) B − Q (cid:107) (cid:46) ,etc., provided thethreshold ε ∗ ∼ for α is sufficiently small.Similar bounds holdfor the adjoints, e.g., (cid:107) ( B − Q ) ∗ (cid:107) (cid:46) . For anillustration how the spectrum of the perturbation B differs from the spectrumof − F , see Figure 8.1.Setting β (0) = 1 − (cid:107) F (cid:107) L → L and b (0) = f , the formulas (8.10b) and (8.10c)amount to determining the subleading order terms of β = β (0) + β (1) α + β (2) α + O ( α + η ) b = b (0) + b (1) α + O B ( α + η ) , (8.22)using the standard perturbation formulas. Writing (8.10a) as B = B (0) + α B (1) + α B (2) + O B → B ( α + η ) , with B (0) = 1 − F , B (1) = − pf , B (2) := − f , we obtain β (1) = (cid:10) b (0) , B (1) b (0) (cid:11) = − i2 (cid:104) pf (cid:105) ,β (2) = (cid:10) b (0) , B (2) b (0) (cid:11) − (cid:68) b (0) , B (1) Q (0) ( B (0) − β (0) ) − Q (0) B (1) b (0) (cid:69) = 2 (cid:0) (cid:107) F (cid:107) L → L (cid:1) (cid:68) Q (0) ( pf ) , (1 − F ) − Q (0) ( pf ) (cid:69) − (cid:10) f (cid:11) + O (cid:16) ηα (cid:17) . (8.23)These expressions match (8.10). To get the last expression of β (2) in (8.23) wehave used (cid:107) Q (0) R ( ζ ) Q (0) (cid:107) L → L ∼ , ζ ∈ [ 0 , β (0) ] , and β (0) ∼ η/α , to approxi-mate ( B (0) − β (0) ) − Q (0) = (1 − F ) − Q (0) + O B → B (cid:16) ηα (cid:17) . b (1) = − ( B (0) − β (0) ) − Q (0) B (1) b (0) = i 2 (1 − F ) − Q (0) ( pf ) + O B (cid:16) ηα (cid:17) . In order to see that ψ ≥ , we use (cid:107) F (cid:107) L → L ≤ to estimate (1 + (cid:107) F (cid:107) L → L ) (1 − F ) − ≥ F )2 . This yields the estimate in (8.13).It remains to prove the / -Hölder continuity of the various quantities in thelemma. To this end we write B ( z ) = e − q ( z ) − (cid:98) F ( | m ( z ) | ) , (8.24)where the operator (cid:98) F ( r ) : B → B is defined in (5.23). Since (cid:107) S (cid:107) ≤ (cid:107) S (cid:107) L → B ∼ it is easy to see from (5.23) that the map r (cid:55)→ (cid:98) F ( r ) is uniformly continuouswhen restricted on the domain of arguments r ∈ B satisfying c/ Φ ≤ r x ≤ Φ .Furthermore, the exponent e − i2 q = ( | m | /m ) , has the same regularity as m because | m | ∼ . Since m ( z ) is uniformly / -Hölder continuous in z (cf. (7.4))we thus have (cid:107) B ( z (cid:48) ) − B ( z ) (cid:107) (cid:46) | z (cid:48) − z | / , (8.25)for any sufficiently close points z and z (cid:48) . The resolvent ( B ( z ) − ζ ) − inheritsthis regularity in z .The continuity of β ( z ) , b ( z ) , P ( z ) in z is proven by representing them ascontour integrals of the resolvent ( B ( z ) − ζ ) − around a contour enclosing theisolated eigenvalue β ( z ) . The functions σ and ψ inherit the / -Hölder regularityfrom their building blocks, − (cid:107) F (cid:107) L → L , f , Q (0) , and the function p . Thecontinuity of the first three follows similarly as that of β , b and Q , using thecontinuity of the resolvent of − F ( z ) in z . Also the continuity of the largesteigenvalue λ ( z ) of F ( z ) is proven this way. In particular, we see from (8.18)that the limit η/α ( z ) exists as z approaches the real line.The function z (cid:55)→ p ( z ) = sign Re m ( z ) , on the other hand, is handled differ-ently. We show that if ε ∗ > sufficiently small, then the restriction of p to aconnected component J of the set { z : α ( z ) ≤ ε ∗ } is a constant, i.e., p ( z (cid:48) ) = p ( z ) ,for any z, z (cid:48) ∈ J . Indeed, since inf x | m x ( z ) | ≥ c , and sup x Im m x ( z ) ≤ C ε ∗ ,for some c , C ∼ , we get (Re m x ) = | m x | − (Im m x ) ≥ c − ( C ε ∗ ) , ∀ x ∈ X . (8.26)Clearly, for a sufficiently small ε ∗ the real part Re m x ( z ) cannot vanish. Con-sequently, the continuity of m : H → B means that the components p x ( z ) =sign Re m x ( z ) ∈ {− , +1 } , may change values only when α ( z ) > ε ∗ .The explicit representation (8.8) of the spectral projector P follows froman elementary property of compact integral operators: If the integral kernel75 T ∗ ) xy of the Hilbert space adjoint of an operator T : L → L , defined by ( T w ) y = (cid:82) T xy w y π (d y ) , has the symmetry ( T ∗ ) xy = T xy , then the right andleft eigenvectors v and v (cid:48) corresponding to the right and left eigenvalues λ and λ ,respectively, are also related by the simple component wise complex conjugation: ( v (cid:48) ) x = v x . We are now ready to show that the projection of u in the b -direction satisfies acubic equation (up to the leading order) provided α and η are sufficiently small.Recall, that T ∗ denotes the L -adjoint of a linear operator T on L . Proposition 8.2 (General cubic equation) . Suppose g ∈ B solves the perturbedQVE (5.35) at z ∈ H with | z | ≤ . Set u := g − m | m | , (8.27a) and define Θ ∈ C and r ∈ B by Θ := (cid:104) b, u (cid:105)(cid:104) b (cid:105) and r := Qu . (8.27b)
There exists ε ∗ ∼ such that if (cid:104) v (cid:105) ≤ ε ∗ , and (cid:107) g − m (cid:107) ≤ ε ∗ , (8.28) then the following holds: The component r is controlled by d and Θ , r = Rd + O B (cid:0) | Θ | + (cid:107) d (cid:107) (cid:1) , (8.29) where R = R ( z ) denotes the bounded linear operator w (cid:55)→ B − Q ( | m | w ) satis-fying (cid:107) R (cid:107) + (cid:107) R ∗ (cid:107) ∼ . (8.30) The coefficient Θ in (8.27) is a root of the complex cubic polynomial, µ Θ + µ Θ + µ Θ + (cid:104)| m | b, d (cid:105) = κ ( u, d ) , (8.31) perturbed by the function κ ( u, d ) of sub-leading order. This perturbation satisfies | κ ( u, d ) | (cid:46) | Θ | + (cid:107) d (cid:107) + | Θ | |(cid:104) e, d (cid:105)| , (8.32) where e : H → B is a uniformly bounded function, (cid:107) e ( z ) (cid:107) (cid:46) , determined by S and a . The coefficient functions µ k : H → C are determined by S and a as ell. They satisfy µ := (cid:16) − (cid:104) f | m |(cid:105) ηα (cid:17) ψ + O ( α ) (8.33a) µ := (cid:16) − (cid:104) f | m |(cid:105) ηα (cid:17) σ + i (3 ψ − σ ) α + O (cid:0) α + η (cid:1) (8.33b) µ := − (cid:104) f | m |(cid:105) ηα + i2 σ α − ψ − σ ) α + O (cid:0) α + η (cid:1) . (8.33c) If z ∈ R , then the ratio η/α is defined through its limit as η → .Finally, the cubic is stable in the sense that | µ ( z ) | + | µ ( z ) | ∼ . (8.34)Note that from (8.27b) and (8.8) we see that Θ is just the component of u in the one-dimensional subspace spanned by b , i.e, P u = Θ b . From (8.27)and (8.14) we read that | Θ | ≤ C ε ∗ is a small parameter along with α and η .Therefore we needed to expand µ to a higher order than µ , which is in turnexpanded to a higher order than µ in the variables α and η in (8.33). Proof.
The proof is split into two separate parts. First, we derive formulas forthe µ k ’s in terms of B, β and b (cf. (8.44) below). Second, we use the formulas(8.10) from Lemma 8.1 to expand µ k ’s further in α and η .First, we write the equation (5.37) in the form Bu = A ( u, u ) + | m | (1 + e − i q u ) d , (8.35)where q = q ( z ) := arg m ( z ) , and the symmetric bilinear map A : B → B , isdefined by A x ( h, w ) := e − i q x (cid:0) h x ( F w ) x + ( F h ) x w x (cid:1) . Clearly, (cid:107)A ( h, w ) (cid:107) (cid:46) (cid:107) h (cid:107)(cid:107) w (cid:107) , since (cid:107) F (cid:107) ≤ (cid:107) m (cid:107) (cid:46) . Applying Q on (8.35)gives r = B − Q A ( u, u ) + B − Q (cid:2) | m | (1 + e − i q u ) d (cid:3) . (8.36)From Lemma 8.1 we know that (cid:107) QB − Q (cid:107) (cid:46) , and hence the boundedness of A implies: (cid:107) B − Q A ( u, u ) (cid:107) (cid:46) (cid:107) u (cid:107) (cid:46) | Θ | + (cid:107) r (cid:107) . From the boundedness of the projections (8.14) (cid:107) r (cid:107) = (cid:107) Qu (cid:107) (cid:46) (cid:107) u (cid:107) ≤ (cid:107) g − m (cid:107) inf x | m x | (cid:46) ε ∗ , where in the second to last inequality we have used | m | ∼ . Plugging this backinto (8.36), we find (cid:107) r (cid:107) ≤ C ( | Θ | + ε ∗ (cid:107) r (cid:107) + (cid:107) d (cid:107) ) , C ∼ . Now we require ε ∗ to be so small that C ε ∗ ≤ , and get (cid:107) r (cid:107) (cid:46) | Θ | + (cid:107) d (cid:107) . (8.37)Applying this on the right hand side of u = Θ b + r yields a uniform bound on u , (cid:107) u (cid:107) (cid:46) | Θ | + (cid:107) d (cid:107) . (8.38)Using the bilinearity and the symmetry of A we decompose r into three parts r = B − Q A ( b, b ) Θ + Rd + (cid:101) r , (8.39)where we have identified the operator R from (8.29), and introduced the sub-leading order part, (cid:101) r := 2 B − Q A ( b, r ) Θ + B − Q A ( r, r ) + B − Q ( | m | e − i q ud )= O B (cid:16) | Θ | + | Θ |(cid:107) d (cid:107) + (cid:107) d (cid:107) (cid:17) . (8.40)Applying the last estimate in (8.39) yields (8.29). We know that B − Q isbounded as an operator on B from (8.9). A direct calculation using (8.8) showsthat also its L -Hilbert-space adjoint satisfies a similar bound, (cid:107) ( B − Q ) ∗ (cid:107) (cid:46) .From this and (cid:107) m (cid:107) (cid:46) the bound (8.30) follows.From (8.8) we see that applying (cid:104) b, · (cid:105) to (8.35) corresponds to projectingonto the b -direction β (cid:104) b (cid:105) Θ= (cid:104) b, A ( b, b ) (cid:105) Θ + 2 (cid:104) b, A ( b, r ) (cid:105) Θ + (cid:104) b, A ( r, r ) (cid:105) + (cid:10) b, | m | (1 + e − i q u ) d (cid:11) = (cid:104) b A ( b, b ) (cid:105) Θ + 2 (cid:10) b A ( b, B − Q A ( b, b )) (cid:11) Θ + (cid:104) b | m | d (cid:105) + κ ( u, d ) , (8.41)where the cubic term corresponds to the part B − Q A ( b, b ) Θ of r in (8.39),while the other parts of (cid:104) b, A ( b, r ) (cid:105) Θ , have been absorbed into the remainderterm, alongside other small terms: κ ( u, d ) := 2 (cid:10) b A ( b, Rd + (cid:101) r ) (cid:11) Θ + (cid:104) b A ( r, r ) (cid:105) + (cid:10) b | m | e − i q ud (cid:11) = (cid:104) e, d (cid:105) Θ + O (cid:0) | Θ | + (cid:107) d (cid:107) (cid:1) , (8.42)where in the second line we have defined e ∈ B in (8.32) such that (cid:104) e, w (cid:105) := 2 (cid:104) b A ( b, Rw ) (cid:105) + (cid:104) b | m | e − i q w (cid:105) , ∀ w ∈ L . For the error estimate in (8.42) we have also used (8.37), (8.38), and (cid:107) b (cid:107) ∼ .This completes the proof of (8.32).From the definitions of A , B , b and β , it follows A ( b, b ) = e − i q b F b = e − i q b (e − i2 q − B ) b = (e − i3 q − β e − i q ) b A ( b, w ) = e − i q (cid:0) b F w − w (e − i2 q − β ) b (cid:1) = b e − i q (e − i2 q + F − β ) w . (8.43) 78sing these formulas in (8.41) we see that the cubic (8.31) holds with the coef-ficients, µ = (cid:68) b e − i q (e − i2 q + F − β ) B − Q (cid:2) b e − i q (e − i2 q − β ) (cid:3)(cid:69) (8.44a) µ = (cid:10) (e − i3 q − β e − i q ) b (cid:11) (8.44b) µ = − β (cid:104) b (cid:105) (8.44c)that are determined by S and z alone.The final expressions (8.33) follow from these formulas by expanding B, β and b , w.r.t. the small parameters α and η using the expansions (8.10). Let uswrite w := (1 − F ) − Q (0) ( pf ) , so that b = f + (i2 w ) α + O B ( α + η ) , and (cid:104) f, w (cid:105) = 0 . Using (5.50) and (5.53)we also obtain an useful representation e − i q = p − i f α + O B ( α + η ) .First we expand the coefficient µ . Using (cid:104) f (cid:105) = 1 and (cid:104) f, w (cid:105) = 0 we obtain (cid:104) b (cid:105) = 1 + O ( α + η ) . Hence, only the expansion of β contributes at the levelof desired accuracy to µ , µ = − β (cid:104) b (cid:105) = − β + O ( α + η )= −(cid:104) f | m |(cid:105) ηα + i2 σα − ψ − σ ) α + O ( α + η ) . Now we expand the second coefficient, µ . Let us first write µ = (cid:10) (e − i3 q − β e − i q ) b (cid:11) = (cid:10) (e − i q b ) (cid:11) − β (cid:10) e − i q b (cid:11) . (8.45)Using the expansions we see that e − i q b = pf + i(2 pw − f ) α + O B ( α + η ) , andthus, taking this to the third power, we find (e − i q b ) = pf + i3(2 pf w − f ) + O B ( α + η ) . Consequently, (cid:10) (e − i q b ) (cid:11) = (cid:104) pf (cid:105) + i3 (cid:2) (cid:104) pf w (cid:105) − (cid:104) f (cid:105) (cid:3) α + O ( α + η )= σ + i3( ψ − σ ) α + O ( α + η ) . (8.46)In order to obtain expressions in terms of σ and ψ = D ( pf ) , where the bilinearpositive form D is defined in (8.13), we have used (cid:10) pf w (cid:11) = (1 + (cid:107) F (cid:107) L → L ) (cid:10) Q (0) ( pf ) , (1 − F ) − Q (0) ( pf ) (cid:11) + O ( η/α ) , as well as the following consequence of P (0) ( pf ) = σf and (cid:107) f (cid:107) = 1 : (cid:104) f (cid:105) = (cid:107) pf (cid:107) = (cid:107) P (0) ( pf ) (cid:107) + (cid:107) Q (0) ( pf ) (cid:107) = σ + (cid:10) Q (0) ( pf ) , Q (0) ( pf ) (cid:11) . (8.47)The expansion of the last term of (8.45) is easy since only β has to be expandedbeyond the leading order. Indeed, directly from (8.10b) we obtain β (cid:10) e − i q b (cid:11) = (cid:16) (cid:104) f | m |(cid:105) ηα − i2 σα + O ( α + η ) (cid:17)(cid:16) (cid:104) pf (cid:105) + O ( α + η ) (cid:17) = − i2 σ α + (cid:104) f | m |(cid:105) ηα σ + O (cid:0) α + η (cid:1) . µ .Finally, µ , is expanded. By the definitions and the identity (5.20) for (cid:107) F (cid:107) L → L we have e − i2 q + F − β = 2 −(cid:104) f | m |(cid:105) ηα − B + O B → B ( α ) = 1+ (cid:107) F (cid:107) L → L − B + O B → B ( α ) . Recalling (cid:107) B − Q (cid:107) (cid:46) and η (cid:46) α , we thus obtain (e − i2 q + F − β ) B − Q = (1 + (cid:107) F (cid:107) L → L ) B − Q − Q + O B → B ( α ) . (8.48)Directly from the definition (8.8) of P = 1 − Q , we see that Q = Q (0) + O B → B ( α ) .Thus BQ = (1 − F ) Q (0) + O B → B ( α ) . Using the general identity, ( A + D ) − = A − − A − D ( A + D ) − , with A :=(1 − F ) Q (0) and A + D := BQ , yields B − Q = (1 − F ) − Q (0) + O B → B ( α ) , (8.49)since B − Q and (1 − F ) − Q (0) are both O B → B (1) . By applying (8.49) in (8.48)we get (e − i2 q + F − β ) ( QBQ ) − = Q (0) (cid:2) (1+ (cid:107) F (cid:107) L → L ) (1 − F ) − − (cid:3) Q (0) + O B → B ( α ) . Using this in the first formula of µ below yields µ = (cid:68) b e − i q (e − i2 q + F − β ) B − Q (cid:0) b e − i q (e − i2 q − β ) (cid:1)(cid:69) = (cid:16) − (cid:104) f | m |(cid:105) ηα (cid:17)(cid:68) Q (0) ( pf ) , (cid:2) (1 + (cid:107) F (cid:107) L → L ) (1 − F ) − − (cid:3) Q (0) ( pf ) (cid:69) + O ( α ) , which equals the second expression (8.33a) because the first term above is D ( pf ) .Finally, we show that | µ | + | µ | ∼ . From the expansion of µ , we get | µ | = (cid:107) F (cid:107) L → L | σ | + O ( α ) (cid:38) | σ | + O ( α ) . Similarly, we estimate from below | µ | (cid:38) ψ + O ( α ) . Therefore, we find that | µ | + | µ | (cid:38) ψ + σ + O ( α ) . We will now show that ψ + σ (cid:38) , which implies | µ | + | µ | (cid:38) , provided theupper bound ε ∗ of α is small enough. Indeed, from the lower bound (8.13) on thequadratic form D , Gap( F ) ∼ and the identity | σ | = |(cid:104) f, pf (cid:105)| = (cid:107) P (0) ( pf ) (cid:107) we conclude that ψ + σ ≥ Gap( F )2 (cid:107) Q (0) ( pf ) (cid:107) + (cid:107) P (0) ( pf ) (cid:107) (cid:38) (cid:107) pf (cid:107) . (8.50)Since inf x f x ∼ and | p | = 1 it follows that (cid:107) pf (cid:107) ∼ .80 hapter 9 Behavior of generatingdensity where it is small
In this chapter we prove Theorem 2.6. We will assume that S satisfies A1-3 and ||| m ||| R ≤ Φ < ∞ . The model parameters are thus the same ones, (8.1),as in the previous chapter. In particular, we have v x ∼ (cid:104) v (cid:105) , and thus thesupport of the components of the generating densities satisfy supp v x = supp v (cf. Definition 4.3). As we are interested in the generating density Im m | R wewill consider m and all the related quantities as functions on R instead of H or H in this chapter.Consider the domain D ε := (cid:8) τ ∈ supp v : (cid:104) v ( τ ) (cid:105) ≤ ε (cid:9) , ε > . (9.1)Theorem 2.6 amounts to showing that for some sufficiently small ε ∼ , v ( τ ) = (cid:98) v ( τ ) + O B (cid:0) (cid:98) v ( τ ) (cid:1) , τ ∈ D ε , (9.2a)holds, where the leading order part factorizes, (cid:98) v x ( τ ) = v x ( τ ) + h x ( τ ) Ψ( τ − τ ; τ ) , ( x, τ ) ∈ X × R , (9.2b)around any expansion point τ from the set of local minima, M ε := (cid:8) τ ∈ D ε : τ is a local minimum of τ (cid:55)→ (cid:104) v ( τ ) (cid:105) (cid:9) , (9.3)and h x ( τ ) ∼ and Ψ( ω ; τ ) ≥ . We show that the function Ψ( ω ; τ ) deter-mining the shape of ω → (cid:104) v ( τ + ω ) (cid:105) is universal in the sense that it dependson τ ∈ M ε only through a single scalar parameter (cf. (2.20)).Let τ denote one of the minima τ k . We consider m ( τ + ω ) as the solutionof the perturbed QVE (5.35) at z = τ with the scalar perturbation d x ( ω ) := ω , ∀ x ∈ X , (9.4) 81nd apply Proposition 8.2. The leading order behavior of m ( τ + ω ) is determinedby expressing u ( ω ; τ ) := m ( τ + ω ) − m ( τ ) | m ( τ ) | , (9.5)as a sum of its projections, Θ( ω ; τ ) b ( τ ) := P ( τ ) u ( ω ; τ ) and r ( ω ; τ ) := Q ( τ ) u ( ω ; τ ) , (9.6)where P = P ( τ ) is defined in (8.8) and Q ( τ ) = 1 − P ( τ ) . The coefficient Θ( ω ; τ ) is then computed as a root of the cubic equation (8.31) correspondingto the scalar perturbation (9.4); its imaginary part will give Ψ( ω, τ ) . Finally,the part r ( ω ; τ ) is shown to be much smaller than Θ( ω ; τ ) so that it canbe considered as an error term. The next lemma collects necessary informa-tion needed to carry out this analysis rigorously. This lemma has appeared asProposition 6.2 in [AEK16b] in the simpler case when the generating densityvanishes at the expansion point, i.e., v ( τ ) = 0 . Lemma 9.1 (Cubic for shape analysis) . There are two constants ε ∗ , δ ∼ ,such that if τ ∈ supp v and (cid:104) v ( τ ) (cid:105) ≤ ε ∗ , (9.7) holds for some fixed base point τ ∈ supp v , then Θ( ω ) = Θ( ω ; τ ) = (cid:28) b ( τ ) (cid:104) b ( τ ) (cid:105) m ( τ + ω ) − m ( τ ) | m ( τ ) | (cid:29) , (9.8) satisfies the perturbed cubic equation µ Θ( ω ) + µ Θ( ω ) + µ Θ( ω ) + Ξ( ω ) ω = 0 , | ω | ≤ δ . (9.9) The coefficients µ k = µ k ( τ ) ∈ C are independent of ω and have expansions in α µ := ψ + κ α (9.10a) µ := σ + i (3 ψ − σ ) α + κ α (9.10b) µ := i2 σ α − ψ − σ ) α + κ α , (9.10c) and Ξ( ω ) = Ξ( ω ; τ ) ∈ C is close to a real constant: Ξ( ω ) := (cid:104) f | m |(cid:105) ( 1 + κ α + ν ( ω )) . (9.11) The scalars α = (cid:104) f, v/ | m |(cid:105) , σ = (cid:104) f, pf (cid:105) and ψ = D ( pf ) are defined in (5.21) , (8.11) and (8.13) , respectively. They are uniformly / -Hölder continuous func-tions of τ on the connected components of the set (cid:8) τ : (cid:104) v ( τ ) (cid:105) ≤ ε ∗ , | τ | ≤ (cid:9) .The cubic (9.9) is stable (cf. (8.34) ) in the sense that | µ | + | µ | ∼ ψ + σ ∼ . (9.12) 82 oth the rest term r ( ω ) = r ( ω ; τ ) (cf. (9.6) ) and Θ( ω ) are differentiable asfunctions of ω on the domain { ω : (cid:104) v ( τ + ω ) (cid:105) > } , and they satisfy: | Θ( ω ) | (cid:46) min (cid:26) | ω | α , | ω | / (cid:27) (9.13a) (cid:107) r ( ω ) (cid:107) (cid:46) | Θ( ω ) | + | ω | . (9.13b) The constants κ j = κ j ( τ ) ∈ C , j = 0 , , , , and ν ( ω ) = ν ( ω ; τ ) ∈ C in (9.10) and (9.11) satisfy | κ | , . . . , | κ | (cid:46) (9.14a) | ν ( ω ) | (cid:46) | Θ( ω ) | + | ω | (cid:46) | ω | / , (9.14b) and ν ( ω ) is / -Hölder continuous in ω .The leading behavior of m on [ τ − δ, τ + δ ] is determined by Θ( ω ; τ ) : m x ( τ + ω )= m x ( τ ) + | m x ( τ ) | b x ( τ ) Θ( ω ; τ ) + O (cid:16) Θ( ω ; τ ) + | ω | (cid:17) (9.15a) = m x ( τ ) + | m x ( τ ) | f x ( τ ) Θ( ω ; τ ) + O (cid:16) α ( τ ) | ω | / + | ω | / (cid:17) . (9.15b) All comparison relations hold w.r.t. the model parameters (8.1) . The expansion (2.19) will be obtained by studying the imaginary parts of(9.15). The factorization (9.2b) corresponds to the factorization of the secondterms on the right hand side of (9.15). In particular, Ψ( ω ; τ k ) = Im Θ( ω ; τ k ) .The universality of the function Ψ( ω ; τ k ) corresponds to Θ( ω ) being close to thesolution of the ideal cubic obtained from (9.9) and by setting κ = κ = κ = 0 and κ = ν ( ω ) = 0 in (9.10) and (9.11), respectively. Proof of Lemma 9.1.
The present lemma is an application of Proposition 8.2in the case where z = τ ∈ supp v and the perturbation is a real number,(9.4). Then the solution to (5.35) is g = m ( τ + ω ) . As for the assumptions ofProposition 8.2, we need to verify the second inequality of (8.28), i.e., (cid:107) m ( τ + ω ) − m ( τ ) (cid:107) ≤ ε ∗ , | ω | ≤ δ . This follows from the uniform / -Hölder continuity of the solution of the QVE(cf. Theorem 2.4) provided we choose δ ∼ ε ∗ sufficiently small. By Theorem 2.4the solution m is also smooth on the set where α > . By Lemma 8.1 and (8.14)the projectors P and Q are uniformly bounded on the connected components ofthe set where α ≤ ε ∗ . This boundedness extends to the real line. Since | m | ∼ ,the functions u ( ω ) and r ( ω ) have the same regularity in ω as m ( τ ) has in τ . Inparticular, (9.13a) follows this way (cf. Corollary 7.3) using α = α ( τ ) ∼ v ( τ ) .Lemma 8.1 implies the Hölder regularity of α, σ, ψ . The estimate (9.12) followsfrom (8.34) provided ε ∗ is sufficiently small. The a priori bound (9.13b) for r follows from the analogous general estimate (8.29).83he formulas (9.10) for the coefficients µ k follow from the general formulas(8.33) by letting η = Im z go to zero. The only non-trivial part is to establish lim η → ηα ( τ + i η ) = 0 , ∀ τ ∈ supp v . (9.16)Since m ( z ) ∈ B is continuous in z , F ( z ) is also continuous as an operator on L . Thus taking the limit Im z → of the identity (4.35) yields v | m | = F v | m | , since | m | ∼ . If Re z = τ , with v ( τ ) (cid:54) = 0 , then the vector v ( τ ) / | m ( τ ) | ∈ L is non-zero, and thus an eigenvector of F corresponding to the eigenvalue . Inparticular, we get (cid:107) F ( τ ) (cid:107) L → L = 1 , τ ∈ supp v . (9.17)If τ ∈ supp v is such that v ( τ ) = 0 then (9.17) follows from a limiting argument τ → τ , with v ( τ ) (cid:54) = 0 , and the continuity of F . Comparing (9.17) with (5.20)implies (9.16).The cubic equation (9.9) in Θ is a rewriting of (8.31). In particular, we have κ α + ν ( ω ) = Ξ( ω ) (cid:104)| m | f (cid:105) = 1 + (cid:104)| m | ( b − f ) (cid:105)(cid:104)| m | f (cid:105) + 1 (cid:104)| m | f (cid:105) κ ( u ( ω ) , ω ) ω , (9.18)where κ ( u, d ) is from (8.31). We set the ω -independent term κ α equal to thesecond term on the right hand side of (9.18). We set ν ( ω ) equal to the last termin (9.18). Clearly, | κ | (cid:46) because b = f + O B ( α ) and | m | , f ∼ . The bound(8.32) and the Hölder continuity of Θ yield (cid:12)(cid:12)(cid:12)(cid:12) κ ( u ( ω ) , ω ) ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) | Θ( ω ) | + | ω || Θ( ω ) | + | ω | | ω | (cid:46) | Θ( ω ) | + | ω | (cid:46) | ω | / . This proves (9.14b). The expansions (9.15) follow by expressing m ( τ + ω ) interms of Θ( ω ; τ ) and r ( ω ; τ ) , and approximating the latter with (8.29).The following ratio, Π( τ ) := | σ ( τ ) |(cid:104) v ( τ ) (cid:105) , (9.19)will play a key role in the classification of the points in D ε when ε > is small.Indeed, the next result shows that if Π is sufficiently large, then v grows at leastlike a square root in the direction sign σ . Lemma 9.2 (Monotonicity) . There exist thresholds ε ∗ , Π ∗ ∼ , such that (sign σ ( τ )) ∂ τ v ( τ ) (cid:38) { Π( τ ) ≥ Π ∗ }| σ ( τ ) | (cid:104) v ( τ ) (cid:105) + (cid:104) v ( τ ) (cid:105) , τ ∈ D ε ∗ . (9.20) 84 roof. By Lemma 9.1 both Θ( ω ; τ ) and r ( ω ; τ ) are differentiable functions in ω , and thus, ∂ τ m ( τ ) = | m ( τ ) | b ( τ ) ∂ ω Θ(0; τ ) + | m ( τ ) | ∂ ω r (0; τ ) . (9.21)Let us drop the fixed argument τ to simplify notations. Taking imaginary partsof (9.21) yields ∂ τ v = Im ∂ τ m = | m | Im (cid:2) b ∂ ω Θ(0) (cid:3) + | m | Im ∂ ω r (0) . (9.22)By dividing (9.13b) by ω , and using (9.13a), we see that (cid:12)(cid:12)(cid:12) r x ( ω ) ω (cid:12)(cid:12)(cid:12) (cid:46) (cid:12)(cid:12)(cid:12) Θ( ω ) ω (cid:12)(cid:12)(cid:12) (cid:46) | ω | α , ∀ x ∈ X . Letting ω → , and recalling r (0) = 0 , we see that the last term in (9.22) isuniformly bounded, (cid:107) Im ∂ ω r (0) (cid:107) (cid:46) . (9.23)We will now show that Im[ b ∂ ω Θ ] dominates the second term in (9.21),provided α is sufficiently small and | σ | /α ∼ Π is sufficiently large. To this endwe first rewrite the cubic (9.9), (cid:18) µ µ Θ( ω ) + µ µ Θ( ω ) (cid:19) Θ( ω ) ω = − Ξ( ω ) µ . (9.24)From the definition (9.10c) we obtain | µ | ∼ α (cid:12)(cid:12) σ + O ( α ) (cid:12)(cid:12) + α (cid:12)(cid:12) ψ − σ + O ( α ) (cid:12)(cid:12) , by distinguishing the cases σ ≤ ψ and σ > ψ , and using (9.12). Applying(9.14b) to estimate Ξ( ω ) we see that the right hand side of (9.24) satisfies Ξ( ω ) µ = (cid:104) f | m |(cid:105) O ( α + | ω | / )i ασ − α ( ψ − σ ) + O ( α ) . (9.25)From (9.13a) we see that Θ( ω ) → as ω → . Hence taking the limit ω → in(9.24) and recalling | µ | , | µ | (cid:46) , yields ∂ ω Θ(0) = dΘd ω (cid:12)(cid:12)(cid:12)(cid:12) ω =0 = (cid:104) f | m |(cid:105) α ( ψ − σ ) + i α σ + O ( α + | σ | α ) α | σ + O ( α ) | + α | ψ − σ + O ( α ) | . (9.26)Using b = f + O B ( α ) and (cid:104) f | m |(cid:105) ∼ , we conclude from (9.26) that (sign σ ) Im (cid:2) b ∂ ω Θ(0) (cid:3) ∼ | σ | + O B ( α + | σ | α ) | σ + O ( α ) | + α | ψ − σ + O ( α ) | α . (9.27)By definitions | σ | /α ∼ Π ≥ Π ∗ . Hence, if Π ∗ ∼ is sufficiently large,then the factor multiplying /α on the right hand side of (9.27) scales like85 in (cid:8) | σ | − , α − | σ | (cid:9) . Here we used again (9.12). Using (9.22), (9.23), and α ∼ (cid:104) v (cid:105) from (9.27) we obtain (sign σ ) ∂ τ v (cid:38) min (cid:26) | σ | , | σ |(cid:104) v (cid:105) (cid:27) (cid:104) v (cid:105) + O B (1) . By taking Π ∗ ∼ sufficiently large and ε ∗ ∼ sufficiently small the term O B (1) can be ignored and (9.20) follows. Lemma 9.2 shows that if τ ∈ D ε ∗ is a non-zero minimum of τ (cid:55)→ (cid:104) v ( τ ) (cid:105) , i.e., (cid:104) v ( τ ) (cid:105) > , then ∂ τ (cid:104) v ( τ ) (cid:105) = 0 , and hence Π( τ ) < Π ∗ . Now we show that anypoint τ satisfying Π( τ ) < Π ∗ is an approximate minimum of (cid:104) v (cid:105) , and its shapeis described by the universal shape function Ψ min : [0 , ∞ ) → [0 , ∞ ) introducedin Definition 2.5. Proposition 9.3 (Non-zero local minimum) . If τ ∈ D ε satisfies Π( τ ) ≤ Π ∗ , (9.28) where Π ∗ ∼ is from Lemma 9.2 ( in particular if τ is a non-zero local minimumof (cid:104) v (cid:105) ) , then v x ( τ + ω ) − v x ( τ ) = h x (cid:104) v (cid:105) Ψ min (cid:18) Γ ω (cid:104) v (cid:105) (cid:19) + O (cid:18) min (cid:26) | ω |(cid:104) v (cid:105) , | ω | / (cid:27)(cid:19) (9.29) for some ω -independent constants h x = h x ( τ ) ∼ and Γ = Γ( τ ) ∼ . Here (cid:104) v (cid:105) = (cid:104) v ( τ ) (cid:105) , σ = σ ( τ ) , etc. are evaluated at τ . Using (2.14b) we see that the first term on the right hand side of (9.29)satisfies (cid:104) v (cid:105) Ψ min (cid:18) Γ ω (cid:104) v (cid:105) (cid:19) ∼ min (cid:26) | ω | (cid:104) v (cid:105) , | ω | / (cid:27) , ω ∈ R . (9.30)Comparing this with the last term of (9.29) we see that the first term dominatesthe error on the right hand side of (9.29) provided (cid:104) v (cid:105) (cid:46) | ω | (cid:46) . Applyingthe lemma at two distinct base points hence yields the following property of thenon-zero minima. Corollary 9.4 (Location of non-zero minima) . Suppose two points τ , τ ∈ D ε satisfy the hypotheses of Proposition 9.3. Then, either | τ − τ | (cid:38) , or | τ − τ | (cid:46) min (cid:8) (cid:104) v ( τ ) (cid:105) , (cid:104) v ( τ ) (cid:105) (cid:9) . (9.31) 86 roof. Suppose the points τ and τ qualify as the base points for Proposi-tion 9.3. Then the corresponding expansions (9.29) are compatible only if thebase points satisfy the dichotomy (9.31). For the second bound in (9.31) we use(9.30).We will use the standard convention on complex powers. Definition 9.5 (Complex powers) . We define complex powers ζ (cid:55)→ ζ γ , γ ∈ C ,on C \ ( −∞ , , by setting ζ γ := exp( γ log ζ ) , where log : C \ ( −∞ , → C isa continuous branch of the complex logarithm with log 1 = 0 . We denote by arg : C \{ } → ( − π, π ) , the corresponding angle function. Proof of Proposition 9.3.
Without loss of generality it suffices to prove(9.29) in the case | ω | ≤ δ for some sufficiently small constant δ ∼ . Indeed,when | ω | (cid:38) the expansion (9.29) becomes trivial since the last term is O (1) and therefore dominates all the other terms, including | v x ( τ ) | ≤ ||| m ||| R ∼ .Similarly, we may restrict ourselves to the setting where the quantity χ := α + | σ | α , (9.32)satisfies χ ≤ χ ∗ , for some sufficiently small threshold χ ∗ ∼ . In particular, weassume that χ ∗ is so small that χ ≤ χ ∗ implies (cid:104) v (cid:105) ≤ ε ∗ .Let us denote by γ k ∈ C , k = 1 , , , ... , generic ω -independent numbers,satisfying | γ k | (cid:46) χ . (9.33)Since Π ∼ | σ | /α and Π ≤ Π ∗ we have | σ | ≤ Π ∗ χ ∗ . From (9.12) it hence followsthat ψ ∼ for sufficiently small χ ∗ ∼ . Thus the cubic (9.9) takes the form Θ( ω ) + i3 α (1 + γ )Θ( ω ) − α (1 + γ )Θ( ω )+ (1 + γ + ν ( ω )) (cid:104) f | m |(cid:105) ψ ω = 0 . (9.34)Using the following normal coordinates , λ := Γ ωα Ω( λ ) := √ (cid:20) (1 + γ ) 1 α Θ (cid:16) α Γ λ (cid:17) + i + γ (cid:21) , (9.35)where Γ := ( √ / (cid:104)| m | f (cid:105) /ψ ∼ , (9.34) reduces to Ω( λ ) + 3 Ω( λ ) + 2Λ( λ ) = 0 . (9.36)Here the constant term Λ : R → C is given by Λ( λ ) := ( 1 + γ + µ ( λ )) λ + γ µ ( λ ) := ν (cid:16) α Γ λ (cid:17) . (9.37) 87he following lemma presents Cardano’s solution for the reduced cubic (9.36)in a form that is convenient for our analysis. We omit the proof of this wellknow result. Lemma 9.6 (Roots of reduced cubic with positive linear coefficient) . The fol-lowing holds: Ω + 3 Ω + 2 ζ = (Ω − (cid:98) Ω + ( ζ ))(Ω − (cid:98) Ω ( ζ ))(Ω − (cid:98) Ω − ( ζ )) , ∀ ζ ∈ C , (9.38) where the three root functions (cid:98) Ω a : C → C , a = 0 , ± , are given by (cid:98) Ω := − odd (cid:98) Ω ± := Φ odd ± i √ even , (9.39a) with Φ even and Φ odd denoting the even and odd parts of the function Φ : C → C , Φ( ζ ) := (cid:0)(cid:112) ζ + ζ (cid:1) / , (9.39b) respectively. The roots (9.39) are analytic and distinct on the set, (cid:98) C := C \{ i ξ : ξ ∈ R , | ξ | > } . (9.40) Indeed, if (cid:98) Ω a ( ζ ) = (cid:98) Ω b ( ζ ) , for a (cid:54) = b , then ζ = ± i . Since Ω( λ ) , defined in (9.35), solves the cubic (9.36), there exists A : R →{ , ±} , such that Ω( λ ) = (cid:98) Ω A ( λ ) (Λ( λ )) , λ ∈ R . (9.41)In the normal coordinates the restriction | ω | ≤ δ becomes | λ | ≤ λ ∗ , where | λ | ≤ λ ∗ := Γ δα . (9.42)Nevertheless, for sufficiently small δ ∼ the function Λ in (9.37) is a smallperturbation of the identity function. Indeed, from (9.37) and the bound (9.14b)on ν , we get | µ ( λ ) | (cid:46) (cid:12)(cid:12)(cid:12) Θ (cid:16) α Γ λ (cid:17)(cid:12)(cid:12)(cid:12) + α | λ | (cid:46) α | λ | / (cid:46) δ / , when | λ | ≤ λ ∗ . (9.43)Hence, if the thresholds δ, χ ∗ (cid:46) are sufficiently small, then Λ( λ ) ∈ G , and | Λ( λ ) | ∼ | λ | , | λ | ≤ λ ∗ , (9.44)where G := (cid:110) ζ ∈ C : dist (cid:0) ζ, i( −∞ , − ∪ i(+1 , + ∞ ) (cid:1) ≥ / (cid:111) . (9.45) 88y Lemma 9.6 the root functions have uniformly bounded derivatives on thissubset of (cid:98) C .The following lemma which is proven in Appendix A.6 is used for replacing Λ( λ ) by λ in (9.41). Lemma 9.7 (Stability of roots) . There exist positive constants c , C such thatif ζ ∈ G and ξ ∈ C satisfy | ξ | ≤ c (1 + | ζ | ) , (9.46) then the roots (9.39) are stable in the sense that (cid:12)(cid:12) (cid:98) Ω a ( ζ + ξ ) − (cid:98) Ω a ( ζ ) (cid:12)(cid:12) ≤ C | ξ | | ζ | / , a = 0 , ± . (9.47)From (9.44) we see that Λ( λ ) (cid:54) = ± i , and hence the roots do not coincide.Moreover, we know from Lemma 9.1 and (9.35): SP-1
The function λ (cid:55)→ Ω( λ ) is continuous.This simple fact will be the first of the four selection principles ( SP ) used fordetermining the correct roots of the cubic (9.9) in the following (cf. Lemma 9.9).Since the roots (cid:98) Ω a | G are also continuous by Lemma 9.6, we conclude that thelabelling function A in (9.41) stays constant on the interval [ − λ ∗ , λ ∗ ] . In orderto determine this constant, a := A ( λ ) , we use the second selection principle: SP-2
The initial value
Ω(0) is consistent with
Θ(0) = 0 .Plugging
Θ(0) = 0 into (9.35) yields
Ω(0) = i √ γ ) = i √ O (cid:16) α + | σ | α (cid:17) . (9.48)On the other hand, using Lemma 9.7 and (9.37) we get (cid:98) Ω a (Λ(0)) = (cid:98) Ω a ( γ ) = (cid:98) Ω a (0) + O (cid:16) α + | σ | α (cid:17) , (9.49)where (cid:98) Ω (0) = 0 and (cid:98) Ω ± (0) = ± i √ . Comparing this with (9.48) and (9.49), we see that for sufficiently small α + | σ | /α (cid:46) χ ∗ , only the the choice A (0) = + satisfies SP-2 .As the last step we derive the expansion (9.29) using the formula v x ( τ + ω ) − v x ( τ ) = | m x | f x Im Θ( ω ) + O (cid:16) α | Θ( ω ) | + | Θ( ω ) | + | ω | (cid:17) , (9.50) 89hich follows by taking the imaginary part of (9.15a). We also used b x = (1 + O ( α )) f x and f x ∼ here. Let us express Θ in terms of the normal coordinatesusing (9.35) Θ( ω ) = α γ (cid:34) (cid:98) Ω + (Λ( λ )) √ − i − γ (cid:35) . (9.51)Here, ω and λ are related by (9.35). Since Θ(0) = 0 , and
Λ(0) = γ (cf. (9.37)),we get i + γ = (cid:98) Ω + ( γ ) √ . Using this identity and Λ( λ ) = γ + Λ ( λ ) with Λ ( λ ) := (1 + γ + µ ( λ )) λ , we rewrite the formula (9.51) as Θ( ω ) = (1 + O ( χ )) α √ (cid:20) (cid:98) Ω + ( γ + Λ ( λ )) − (cid:98) Ω + ( γ ) (cid:21) . (9.52)From (9.44) we know that the arguments of (cid:98) Ω + in (9.52) are in G . Using theuniform boundedness of the derivatives of Ω | G , and the bound | Φ( ζ ) | (cid:46) | ζ | / ,we get (cid:12)(cid:12) (cid:98) Ω + ( γ + Λ ( λ )) − (cid:98) Ω + ( γ ) (cid:12)(cid:12) (cid:46) min (cid:8) | λ | , | λ | / (cid:9) , | λ | ≤ λ ∗ . (9.53)By using (9.53) in (9.43) and (9.52) we estimate the sizes of both µ ( λ ) and Θ( ω ) , | µ ( λ ) | + (cid:12)(cid:12)(cid:12) Θ (cid:16) α Γ λ (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) α min (cid:8) | λ | , | λ | / (cid:9) , | λ | ≤ λ ∗ . (9.54)In order to extract the exact leading order terms, we express the differenceon the right hand side of (9.52) using the mean value theorem (cid:98) Ω + ( γ + Λ ( λ )) − (cid:98) Ω + ( γ ) = (cid:98) Ω + (Λ ( λ )) − (cid:98) Ω + (0)+ γ ∂∂ζ (cid:104) (cid:98) Ω + ( ζ + Λ ( λ )) − (cid:98) Ω + ( ζ ) (cid:105) ζ = γ , (9.55)where γ ∈ G is some point on the line segment connecting and γ . Using(9.54) and Lemma 9.7 on the first term on the right hand side of (9.55) shows (cid:98) Ω + (Λ ( λ )) − (cid:98) Ω + (0) = (cid:98) Ω + ( λ ) − (cid:98) Ω + (0) + O (cid:16) χ min (cid:8) | λ | , | λ | / (cid:9)(cid:17) . (9.56)From an explicit calculation we get | ∂ ζ (cid:98) Ω + ( ζ ) | (cid:46) , for ζ ∈ G . Thus (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ζ (cid:104) (cid:98) Ω + ( ζ + Λ ( λ )) − (cid:98) Ω + ( ζ ) (cid:105) ζ = γ (cid:12)(cid:12)(cid:12)(cid:12) (cid:46) min (cid:8) | λ | , (cid:9) . (cid:98) Ω + ( γ + Λ ( λ )) − (cid:98) Ω + ( γ ) = (cid:98) Ω + ( λ ) − (cid:98) Ω + (0) + O (cid:16) χ min (cid:8) | λ | , | λ | / (cid:9)(cid:17) . (9.57)Via (9.52) we use this to represent the leading order term in (9.50). By approx-imating all the other terms in (9.50) with (9.54) we obtain v x ( τ + ω ) − v x ( τ )= | m | x f x α Im (cid:2) (cid:98) Ω + ( λ ) − (cid:98) Ω + (0) (cid:3) √ O (cid:18)(cid:0) α + | σ | (cid:1) min (cid:8) | λ | , | λ | / (cid:9)(cid:19) . (9.58)Using the formulas (9.39) and (9.39b), we identify the universal shape functionfrom (2.14b), Ψ min ( λ ) = Im (cid:2) (cid:98) Ω + ( λ ) − (cid:98) Ω + (0) (cid:3) √ . Denoting h x := ( α/ (cid:104) v (cid:105) ) f x and writing λ in terms of ω in (9.58) the expansion(9.29) follows. Together with Proposition 9.3 the next result covers the behavior of v | D ε aroundits minima for sufficiently small ε ∼ . For each τ ∈ ∂ supp v , satisfying σ ( τ ) (cid:54) = 0 , we associate the gap length , ∆( τ ) := inf (cid:8) ξ ∈ (0 ,
2Σ ] : (cid:10) v ( τ − sign σ ( τ ) ξ ) (cid:11) > (cid:9) , (9.59)with the convention ∆( τ ) := 2Σ in case the infimum does not exist. We willsee below that if τ ∈ ∂ supp v , then σ ( τ ) (cid:54) = 0 and sign σ ( τ ) is indeed thedirection in which the set supp v continues from τ . Because supp v ⊂ [ − Σ , Σ ] the number ∆( τ ) thus defines the length of the actual gap in supp v startingat τ , with the convention that the gap length is for the extreme edges.Recall the definition (2.14a) of the universal edge shape function Ψ edge :[0 , ∞ ) → [0 , ∞ ) . Proposition 9.8 (Vanishing local minimum) . Suppose τ ∈ supp v with v ( τ ) =0 . Depending on the value of σ = σ ( τ ) either of the following holds: (i) If σ ( τ ) (cid:54) = 0 , then τ ∈ ∂ supp v and supp v continues in the direction sign σ , such that for (sign σ ) ω ≥ , v x ( τ + ω ) = h x ∆ / Ψ edge (cid:18) | ω | ∆ (cid:19) + O (cid:18) min (cid:26) | ω | ∆ / , | ω | / (cid:27)(cid:19) , (9.60) 91 here h x = h x ( τ ) ∼ , and ∆ = ∆( τ ) is the length of the gap in supp v in the direction − sign σ from τ (cf. (9.59) ). Furthermore, the gap lengthsatisfies ∆( τ ) ∼ | σ ( τ ) | , (9.61) while the shapes in the x -direction match at the opposite edges of the gapin the sense that h ( τ ) = h ( τ ) + O B ( ∆ / ) , for τ = τ − sign σ ( τ ) ∆ . (ii) If σ ( τ ) = 0 , then dist( τ , ∂ supp v ) ∼ , and for some h x = h x ( τ ) ∼ : v x ( τ + ω ) = h x | ω | / + O (cid:0) | ω | / (cid:1) . (9.62)From the explicit formula (2.14a) one sees that the leading order term in(9.60) satisfies ∆ / Ψ edge (cid:18) ω ∆ (cid:19) ∼ ω / ∆ / when ≤ ω (cid:46) ∆ ; ω / when ω (cid:38) ∆ . (9.63)In particular, if an edge τ is separated by a gap of length ∆( τ ) ∼ from theopposite edge of the gap, then v grows like a square root.Proposition 9.8 is proven at the end of Subsection 9.2.2 by combining variousauxiliary results which we prove in the following two sections. What is commonwith these intermediate results is that the underlying cubic (9.9) is always ofthe form ψ Θ( ω ) + σ Θ( ω ) + (1 + ν ( ω )) (cid:104)| m | f (cid:105) ω = 0 , ψ + | σ | ∼ , (9.64)since α ( τ ) = v ( τ ) = 0 at the base point τ . In order to analyze (9.64) webring it to a normal form by an affine transformation. This corresponds toexpressing the variables ω and Θ in terms of normal variables Ω and λ , suchthat Ω( λ ) = κ Θ(Γ λ ) + Ω , (9.65)with some λ -independent parameters κ = κ ( τ ) , Γ = Γ( τ ) > , and Ω ∈ C .These parameters will be defined on a case by case basis. We remark, that inthe proof of Proposition 9.3 the coordinate transformations (9.35) were of theform (9.65).In the following, the variable Ω( λ ) will be identified with roots of variouscubic polynomials that depend on the type of base points τ , similarly to (9.41)above. In order to choose the correct roots we use the following selectionprinciples . Lemma 9.9 (Selection principles) . If v ( τ ) = 0 at the base point τ ∈ supp v ofthe expansion (9.65) , then Ω( λ ) = Ω( λ ; τ ) defined in (9.65) has the properties: P-1 λ (cid:55)→ Ω( λ ) is continuous; SP-2
Ω(0) = Ω ; SP-3 Im (cid:2) Ω( λ ) − Ω(0) (cid:3) ≥ , ∀ λ ∈ R ; SP-4
If the imaginary part of Ω grows slower than a square root in a direction θ ∈ {± } , lim ξ → + ξ − / Im Ω( θξ ) = 0 , then Ω | I is real and non-decreasing on an interval I := { θξ : 0 < ξ < ∆ } ,with some ∆ > . For the proof, by combining (9.8), (8.10c) and (9.65) we see that Ω( λ ) = κ (cid:68) f | m | , m ( τ + Γ λ ) − m ( τ ) (cid:69) + Ω , (9.66)where κ, Γ > and Ω ∈ C are from (9.65). Thus the first three selectionprinciples follow trivially from the corresponding properties Im m ( τ ) = 0 and Im m ( τ ) ≥ of m . The property SP-4 follows from (9.66) and the next result.
Lemma 9.10 (Growth condition) . Suppose v ( τ ) = 0 and that (cid:104) v (cid:105) grows slowerthan any square-root in a direction θ ∈ {±} , i.e., lim inf ξ → + (cid:104) v ( τ + θξ ) (cid:105) ξ / = 0 . (9.67) Then (cid:104) v (cid:105) actually vanishes, Im (cid:104) m (cid:105)| I = 0 , while Re (cid:104) m (cid:105) is non-decreasing onsome interval I = { θ ξ : 0 ≤ ξ ≤ ∆ } , for some ∆ > .If the lim inf in (9.67) is non-zero, then either θ = sign σ ( τ ) or σ ( τ ) = 0 . Proof.
We will prove below that if v ( τ ) = 0 , and inf (cid:8) ξ > (cid:104) v ( τ + θξ ) (cid:105) > (cid:9) = 0 (9.68)for some direction θ ∈ {± } , then lim inf ξ → + (cid:104) v ( τ + θξ ) (cid:105) ξ / > . (9.69)Assuming this implication, the lemma follows easily: If (9.67) holds, then(9.68) is not true, i.e., there is a non-trivial interval I = { θξ : 0 ≤ ξ ≤ ∆ } , ∆ > , such that v | I = 0 . As the negative of a Hilbert-transform of v x (cf.(2.8)), the function τ (cid:55)→ Re m x ( τ ) , is non-decreasing on I . This proves the firstpart of the lemma.We will now prove that (9.68) implies (9.69). The key idea is to use Lemma 9.2to prove that (cid:104) v (cid:105) grows at least like a square root. However, first we use Propo-sition 9.3 to argue that the indicator function on the right hand side of (9.20) isnon-zero in a non-trivial neighborhood of τ . To this end, assume < (cid:104) v ( τ ) (cid:105) ≤ ε Π( τ ) < Π ∗ . If ε, δ > are sufficiently small, then Proposition 9.3 can beapplied with τ as the base point. In particular, (9.29) and (9.30) imply (cid:104) v ( τ + ω ) (cid:105) ∼ (cid:104) v ( τ ) (cid:105) + | ω | / > , | ω | ≤ δ . (9.70)Suppose τ satisfies (9.68). Since v ( τ ) = 0 the lower bound in (9.70), appliedto ω = τ − τ , implies | τ − τ | > δ . As τ was arbitrary we conclude Π( τ ) ≥ Π ∗ for every τ in the set I := (cid:8) τ ∈ R : | τ − τ | ≤ δ , < (cid:104) v ( τ ) (cid:105) ≤ ε (cid:9) . Applying Lemma 9.2 on I , recalling the upper bound on | ∂ z m | from Corol-lary 7.3, yields (cid:104) v (cid:105) − (cid:46) (sign σ ) ∂ τ (cid:104) v (cid:105) (cid:46) (cid:104) v (cid:105) − , on I . (9.71)Since v is analytic when non-zero, and dist( τ , I ) = 0 by (9.68), we concludethat I equals the interval with end points τ and τ := τ + θδ . Here we set δ (cid:46) ε so small that the / -Hölder continuity of m guarantees (cid:104) v (cid:105) ≤ ε on I .Moreover, sign σ ( τ ) must equal the constant θ for every τ ∈ I : If σ changed itssign at some point τ ∗ ∈ I this would violate Π( τ ∗ ) ≥ Π ∗ as (cid:104) v (cid:105) is a continuousfunction.Integrating (9.71) from τ to τ we see that (cid:104) v ( τ + θξ ) (cid:105) (cid:38) ξ for any ξ ≤| τ − τ | . This proves the limit (9.69), and hence the first part of the lemma.The second part of the lemma follows from (9.71). When | σ | > and | ω | is sufficiently small compared to | σ | the cubic term ψ Θ( ω ) in (9.64) can be ignored. In this regime the following simple expansion holdsshowing the square root behavior of v near an edge of its support. Lemma 9.11 (Simple edge) . If τ ∈ supp v satisfies v ( τ ) = 0 and σ = σ ( τ ) (cid:54) =0 , then v x ( τ + ω ) = h (cid:48) x (cid:12)(cid:12)(cid:12) ωσ (cid:12)(cid:12)(cid:12) / + O (cid:16) ωσ (cid:17) if ≤ (sign σ ) ω ≤ c ∗ | σ | ;0 if − c ∗ | σ | ≤ (sign σ ) ω ≤ (9.72) for some sufficiently small c ∗ ∼ . Here h (cid:48) = h (cid:48) ( τ ) ∈ B satisfies h (cid:48) x ∼ . This result already shows that supp v continues in the direction sign σ ( τ ) and in the opposite direction there is a gap of length ∆( τ ) (cid:38) | σ ( τ ) | in theset supp v . We will see later (cf. Lemma 9.17) that for small | σ ( τ ) | there isan asymptotically sharp correspondence between ∆( τ ) and | σ ( τ ) | , as ∆( τ ) becomes very small. 94 roof. Treating the cubic term ψ Θ in (9.64) as a perturbation, (9.64) takesthe form Ω( λ ) + Λ( λ ) = 0 , (9.73)in the normal coordinates, λ := ωσ Ω( λ ) := Θ( σλ ) (cid:112) (cid:104)| m | f (cid:105) , (9.74)where Λ : R → C is a multiplicative perturbation of λ : Λ( λ ) := (1 + µ ( λ )) λ µ ( λ ) := 1 + ν ( σλ )1 + ( ψ/σ ) Θ( σλ ) . (9.75)Let λ ∗ = c ∗ | σ | , with some c ∗ ∼ , so that the constraint | ω | ≤ c ∗ | σ | in (9.72)translates into | λ | ≤ λ ∗ .Using the a priori bounds (9.13a) and (9.14b) for Θ and ν yields | µ ( λ ) | (cid:46) (cid:16) ψ | σ | (cid:17)(cid:12)(cid:12) Θ( σλ ) (cid:12)(cid:12) + | σ | | λ | (cid:46) c / ∗ . (9.76)Hence, for sufficiently small c ∗ ∼ we get | µ ( λ ) | < , provided | λ | ≤ λ ∗ .Let us define two root functions (cid:98) Ω a : C → C , a = ± , such that (cid:98) Ω a ( ζ ) + ζ = 0 , (9.77)by setting (cid:98) Ω ± ( ζ ) := ± (cid:40) i ζ / if Re ζ ≥ − ( − ζ ) / if Re ζ < . (9.78)Note that we use the same symbol (cid:98) Ω a for the roots as in (9.39) for differentfunctions. In each expansion (cid:98) Ω a will denote the root function of the appropriatenormal form of the cubic.Comparing (9.73) and (9.77) we see that there exists a labelling function A : R → {±} , such that Ω( λ ) = (cid:98) Ω A ( λ ) (Λ( λ )) , for every λ ∈ R . The function A | [ − λ ∗ ,λ ∗ ] will now be determined using theselection principles SP-1 and
SP-3 .The restrictions of the root functions onto the half spaces Re ζ > and Re ζ < are continuous (analytic) and distinct, i.e., (cid:98) Ω + ( ζ ) (cid:54) = (cid:98) Ω − ( ζ ) for ζ (cid:54) = 0 .Since Ω : R → C is also continuous by SP-1 , A ( λ ) may change its value at some95oint λ = λ only if Λ( λ ) = 0 . Since | µ ( λ ) | < for | λ | ≤ λ ∗ we conclude that Λ( λ ) = 0 only for λ = 0 . Thus, there exist two labels a + , a − ∈ {±} , such that A ( λ ) = a ± ∀ λ ∈ ± [ 0 , λ ∗ ] . (9.79)Let us first consider the case λ ≥ , and show that a + = + . Indeed, thechoice a + = − is ruled out, since Im (cid:98) Ω − (Λ( λ )) = Im (cid:104) − i (1 + µ ( λ )) / λ / (cid:105) = − λ / + O (cid:16) µ ( λ ) λ / (cid:17) (9.80)is negative for sufficiently small c ∗ ∼ in (9.76), and this violates the selectionprinciple SP-3 .By definitions, | Θ( σλ ) | ∼ | (cid:98) Ω + ( λ ) | (cid:46) | Λ( λ ) | / ∼ | λ | / . Using ψ/ | σ | (cid:46) | σ | − , with | σ | (cid:38) , we write (9.76) in the form | µ ( λ ) | (cid:46) | σ | − | λ | / . Similarly, as (9.80) we obtain Ω( λ ) = (cid:98) Ω + ( λ ) + O (cid:16) µ ( λ ) λ / (cid:17) = i λ / + O (cid:16) λσ (cid:17) , λ ∈ [ 0 , λ ∗ ] . Inverting (9.74) we obtain
Im Θ( ω ) = (cid:104)| m | f (cid:105) / (cid:12)(cid:12)(cid:12) ωσ (cid:12)(cid:12)(cid:12) / + O (cid:16) ωσ (cid:17) , sign σ = sign ω . (9.81)Taking the imaginary part of (9.15a) and using (9.81) yields the first line of(9.72), with h (cid:48) x = | m x | f x / (cid:104)| m | f (cid:105) / . Since | m x | , f x ∼ , we also have h (cid:48) x ∼ .In order to prove the second line of (9.72) we show that the gap length (cf.(9.59)) satisfies ∆( τ ) (cid:38) | σ ( τ ) | . (9.82)At the opposite edge of the gap τ := τ − sign σ ( τ ) ∆( τ ) , the density (cid:104) v (cid:105) increases, by definition, in the opposite direction than at τ . By Lemma 9.10 theaverage generating density (cid:104) v (cid:105) increases at least like a square root function andeither sign σ ( τ ) = − sign σ ( τ ) or σ ( τ ) = 0 . Since σ is / -Hölder continuous, σ can not change arbitrarily fast. Namely, we have ∆( τ ) (cid:38) | σ ( τ ) | , and thisproves (9.82).Although not necessary for the proof of the present lemma, it can be shownthat a − := sign σ using the selection principle SP-4 . The same reasoning willbe used in the proofs of the next two lemmas (cf. (9.91) and discussion afterthat).Next we consider the marginal case where the term σ Θ( ω ) is absent in thecubic (9.64). In this case (cid:104) v (cid:105) has a cubic root cusp shape around the base point.96 emma 9.12 (Vanishing quadratic term) . If τ ∈ supp v is such that v ( τ ) = σ ( τ ) = 0 , then v x ( τ + ω ) = h x | ω | / + O (cid:16) | ω | / (cid:17) , (9.83) where h = h ( τ ) ∈ B satisfies h x ∼ . Contrasting this with Lemma 9.11 shows that σ ( τ ) (cid:54) = 0 for τ ∈ ∂ supp v .In particular, the gap length ∆( τ ) is always well defined for τ ∈ ∂ supp v (cf.(9.59)). Proof.
First we note that it suffices to prove (9.83) only for | ω | ≤ δ , where δ ∼ can be chosen to be sufficiently small. When | ω | > δ the last term maydominate the first term on the right hand side of (9.83), and thus we havenothing prove. Since σ = 0 , the quadratic term is missing in (9.64), and thusthe cubic reduces to Ω( ω ) + Λ( ω ) = 0 , (9.84)using the normal coordinates λ := ω Ω( λ ) := (cid:16) ψ (cid:104)| m | f (cid:105) (cid:17) / Θ( λ ) . (9.85)Here, Λ : R → C is a perturbation of the identity function: Λ( λ ) := (1 + ν ( λ )) λ . (9.86)Note that ψ ∼ because of (9.12).Let us define three root functions (cid:98) Ω a : C → C , a = 0 , ± , satisfying (cid:98) Ω a ( ζ ) + ζ = 0 , by the explicit formulas (cid:98) Ω ( ζ ) := − p ( ζ ) (cid:98) Ω ± ( ζ ) := − ± i √ p ( ζ ) , (9.87)where p : C → C is a (non-standard) branch of the complex cubic root, p ( ζ ) := (cid:40) ζ / when Re ζ > − ( − ζ ) / when Re ζ < . (9.88)From (9.84) we see that there exists a labelling A : R → { , ±} , such that Ω( λ ) = (cid:98) Ω A ( ω ) (Λ( λ )) . (9.89) 97imilarly as before, we conclude that Ω and the roots are continuous (cf. SP-1 ) on R and on the half-spaces { ζ ∈ C : ± Re ζ > } , respectively. This impliesthat A ( λ − (cid:54) = A ( λ + 0) if and only if Λ( λ ) = 0 . From the a priori estimate | ν ( λ ) | (cid:46) | λ | / (cf. (9.14b)) we see that there exists δ ∼ such that Λ( λ ) (cid:54) = 0 ,for < | λ | ≤ δ . Hence, we conclude A ( λ ) = a ± , ∀ λ ∈ ± (0 , δ ] . (9.90)The choices a + = − and a − = + are excluded by the selection principle SP-3 : Similarly as in (9.80), we get ± (sign λ ) Im (cid:98) Ω ± (Λ( λ )) = √ | λ | / + O (cid:16) µ ( λ ) λ / (cid:17) ≥ | λ | / − C | λ | / . (9.91)From this it follows that Im (cid:98) Ω − (Λ( λ )) < for small | λ | > . Thus SP-3 implies a ± (cid:54) = ∓ .We will now exclude the choices a ± = 0 . Similarly as (9.91) we use (9.14b)to get Re (cid:98) Ω (Λ( λ )) ≤ − λ / + Cλ / Im (cid:98) Ω (Λ( λ )) (cid:46) | ν ( λ ) | | λ | / (cid:46) | λ | / , (9.92)for λ ≥ . If a + = 0 , then these two bounds together would violate SP-4 . Thechoice a − = 0 is excluded similarly. Thus we are left with the unique choices a + = + and a − = − .The expansion (9.83) is obtained similarly as in the proof of Lemma 9.11.First, we use (9.85) and (9.91) to solve for Im Θ( ω ) . Then we take the imaginarypart of (9.15b) to express v x ( τ + ω ) in terms of Im Θ( ω ) . We identify h x := √ (cid:16) (cid:104)| m | f (cid:105) ψ (cid:17) / | m x | f x , in the expansion (9.83). From ψ, | m | , f ∼ it follows that h x ∼ . In this section we consider the generic case of the cubic (9.64) where neitherthe cubic nor the quadratic term can be neglected. First, we remark thatLemma 9.11 becomes ineffective as | σ | approaches zero since the cubic termof ψ Θ( ω ) + σ Θ( ω ) + (1 + ν ( ω )) (cid:104)| m | f (cid:105) ω = 0 , ψ, σ (cid:54) = 0 , (9.93)was treated as a perturbation of a quadratic equation along with ν ( ω ) in theproof. Thus we need to consider the case where | σ | is small. Indeed, we will as-sume that | σ | ≤ σ ∗ , where σ ∗ ∼ is a threshold parameter that will be adjusted98o that the analysis of the cubic (9.93) simplifies sufficiently. In particular, wewill choose σ ∗ so small that the number (cid:98) ∆ = (cid:98) ∆( τ ) > defined by (cid:98) ∆ := 427 (cid:104)| m | f (cid:105) | σ | ψ , (9.94)satisfies (cid:98) ∆ ∼ | σ | , provided | σ | ≤ σ ∗ . (9.95)Note that the existence of σ ∗ ∼ such that (9.95) holds follows from f x , | m x | ∼ and the stability of the cubic (9.12). Indeed, (9.12) shows that ψ ∼ when | σ | ≤ σ ∗ for some small enough σ ∗ ∼ . We will see below (cf. Lemma 9.17)that (cid:98) ∆( τ ) approximates the gap length ∆( τ ) when the latter is small.Introducing the normal coordinates, λ := 2 ω (cid:98) ∆Ω( λ ) := 3 ψ | σ | Θ (cid:16) (cid:98) ∆2 λ (cid:17) + sign σ , (9.96)the generic cubic (9.93) reduces to Ω( λ ) − λ ) + 2 Λ( λ ) = 0 , (9.97)with the constant term Λ( λ ) := sign σ + (1 + µ ( λ )) λ , (9.98) µ ( λ ) := ν (cid:16) (cid:98) ∆2 λ (cid:17) . (9.99)Here, Λ( λ ) is considered as a perturbation of sign σ + λ . Indeed, from (9.14b)and (9.99) we see that | µ ( λ ) | (cid:46) δ / .The left hand side of equation (9.97) is a cubic polynomial of Ω( λ ) with aconstant term Λ( λ ) . It is very similar to (9.36) but with an opposite sign in thelinear term. Cardano’s formula in this case read as follows. Lemma 9.13 (Roots of reduced cubic with negative linear coefficient) . For any ζ ∈ C , Ω − ζ = (Ω − (cid:98) Ω + ( ζ ))(Ω − (cid:98) Ω ( ζ ))(Ω − (cid:98) Ω − ( ζ )) , (9.100) where the three root functions (cid:98) Ω (cid:36) : C → C , (cid:36) = 0 , ± , have the form (cid:98) Ω := − (Φ + + Φ − ) (cid:98) Ω ± := 12 (Φ + + Φ − ) ± i √
32 (Φ + − Φ − ) . (9.101a) 99 he auxiliary functions Φ ± : C → C , are defined by (recall Definition 9.5) Φ ± ( ζ ) := (cid:0) ζ ± (cid:112) ζ − (cid:1) / if Re ζ ≥ , (cid:0) ζ ± i (cid:112) − ζ (cid:1) / if | Re ζ | < , − (cid:0) − ζ ∓ (cid:112) ζ − (cid:1) / if Re ζ ≤ − . (9.101b) On the simply connected complex domains (cid:98) C := (cid:8) ζ ∈ C : | Re ζ | < (cid:9) , and (cid:98) C ± := (cid:8) ζ ∈ C : ± Re ζ > (cid:9) , (9.102) the respective restrictions of (cid:98) Ω a are analytic and distinct. Indeed, if (cid:98) Ω a ( ζ ) = (cid:98) Ω b ( ζ ) holds for some a (cid:54) = b and ζ ∈ C , then ζ = ± . This lemma is analogue of Lemma 9.6 but for (9.97) instead of (9.36). Asbefore the meaning of the symbols (cid:98) Ω a , λ , etc., is changed accordingly.Comparing (9.97) and (9.100) we see that there exists a function A : R →{ , ±} such that Ω( λ ) = (cid:98) Ω A ( λ ) (Λ( λ )) . (9.103)We will determine the values of A inside the following three intervals I := − (sign σ )[ − λ , I := − (sign σ )( 0 , λ ] I := − (sign σ )[ λ , λ ] , (9.104)which are defined by their boundary points, λ := 2 δ (cid:98) ∆ , λ := 2 − (cid:37) | σ | , λ := 2 + (cid:37) | σ | , (9.105)for some (cid:37) ∼ . The shape of the imaginary parts of the roots (cid:98) Ω a on the intervals I , I and I is shown in Figure 9.1. The number λ is the expansion range δ in the normal coordinates. From (9.95) it follows that c δ | σ | ≤ λ ≤ C δ | σ | , provided | σ | ≤ σ ∗ . (9.106)The points λ and λ will act as a lower and an upper bound for the size ofthe gap in supp v associated to the edge τ , respectively. Given any δ, (cid:37) ∼ wecan choose σ ∗ ∼ so small that λ ≥ , and ≤ λ < < λ ≤ , provided | σ | ≤ σ ∗ . (9.107)In particular, the intervals (9.104) are disjoint and non-trivial for a triple ( δ, (cid:37), σ ∗ ) chosen this way. The value A ( λ ) can be uniquely determined using the selectionprinciples if λ lies inside one of the intervals (9.104).100igure 9.1: Imaginary parts of the three branches of the roots of the cu-bic equation. The true solution remains within the allowed error marginindicated by the dashed lines. Lemma 9.14 (Choice of roots) . There exist δ, (cid:37), σ ∗ ∼ , such that (9.107) holds, and if | σ | ≤ σ ∗ , then the restrictions of Ω on the intervals I k := I k ( δ, (cid:37), σ, (cid:98) ∆) , defined in (9.104) ,satisfy: Ω | I = (cid:98) Ω + ◦ Λ | I Ω | I = (cid:98) Ω − ◦ Λ | I Ω | I = (cid:98) Ω + ◦ Λ | I . (9.108) Moreover, we have
Im Ω( − sign σ λ ) > . (9.109)The proof of the following simple result is given in Appendix A.6. Lemma 9.15 (Stability of roots) . On the connected components of (cid:98) C the roots (9.101a) are stable, i.e., (cid:12)(cid:12) (cid:98) Ω a ( ζ ) − (cid:98) Ω a ( ξ ) (cid:12)(cid:12) (cid:46) min (cid:110) | ζ − ξ | / , | ζ − ξ | / (cid:111) , ( ζ, ξ ) ∈ (cid:98) C − ∪ (cid:98) C ∪ (cid:98) C , (9.110) 101 olds for a = − , , + .In particular, suppose ζ and ξ are of the following special form ξ = − θ + λζ = − θ + (1 + µ (cid:48) ) λ , where θ = ± , λ ∈ R and µ (cid:48) ∈ C . Suppose also that | λ − θ | ≥ κ , and | µ (cid:48) | ≤ κ ,for some κ ∈ (0 , / . Then for each a = − , , + the function (cid:98) Ω a satisfies (cid:12)(cid:12) (cid:98) Ω a ( ζ ) − (cid:98) Ω a ( ξ ) (cid:12)(cid:12) (cid:46) min (cid:8) | λ | / , | λ | / (cid:9) κ / | µ (cid:48) | . (9.111)Using Lemma 9.15 we may treat Λ( λ ) as a perturbation of sign σ + λ bya small error term λ µ ( λ ) . By expressing the a priori bounds (9.14b) for ν ( ω ) in the normal coordinates (9.96), and recalling that | λ | ≤ λ is equivalent to | ω | ≤ δ , we obtain estimates for this error term, | µ ( λ ) | ≤ C | σ || λ | / (9.112a) ≤ C δ / , provided | σ | ≤ σ ∗ , | λ | ≤ λ . (9.112b)In the following we will assume that δ ≤ (2 C ) − ∼ , so that sup λ : | λ |≤ λ | µ ( λ ) | ≤ , provided | σ | ≤ σ ∗ . (9.112c)The a priori bound in the middle of (9.14b) also yields the third estimate of µ in terms of Ω and λ . Indeed, inverting (9.96) and using Ω(0) = sign σ = 1 (alsofrom (9.96)), we get | µ ( λ ) | (cid:46) | σ || Ω( λ ) − Ω(0) | + | σ | | λ | , provided | σ | ≤ σ . (9.112d)For the sake of convenience, we will restrict our analysis to the case sign σ = − .The opposite case is handled similarly.We will use the notations ϕ ( τ + 0) and ϕ ( τ − , for the right and the leftlimits lim ξ ↓ τ ϕ ( ξ ) and lim ξ ↑ τ ϕ ( ξ ) , respectively. Proof of Lemma 9.14.
Let us assume sign σ = − . We will consider δ ∼ and (cid:37) ∼ as free parameters which can be adjusted to be as small and largeas we need, respectively. Given δ ∼ and (cid:37) ∼ the threshold σ ∗ ∼ is thenchosen so small that (9.107) holds.First we show that A ( λ ) is constant on each I k , i.e., there are three labels a k ∈ { , ±} such that A ( λ ) = a k , ∀ λ ∈ I k , k = 1 , , . (9.113)In order to prove this we first recall that the root functions ζ (cid:55)→ (cid:98) Ω a ( ζ ) , a, b =0 , ± , are continuous on the domains (cid:98) C b , b = 0 , ± , and that they may coincide102nly at points Re ζ = ± (Indeed, the roots coincide only at the two points ζ = ± .). From Lemma 9.1 and SP-1 we see that Λ , Ω : R → C are continuous.Hence, (9.113) will follow from Λ( I ) ⊂ (cid:98) C − , Λ( I ) ⊂ (cid:98) C , Λ( I ) ⊂ (cid:98) C + , (9.114)since | Re ζ | (cid:54) = 1 for ζ ∈ ∪ a (cid:98) C a (cf. (9.102)).From (9.98) and (9.112c) we get Re Λ( λ ) = − − (1 + µ ( λ )) | λ | ≤ − − | λ | < − , λ ∈ I , (9.115)and thus Λ( I ) ⊂ (cid:98) C − . Similarly, we get the first estimate below: − | λ | ≤ Re Λ( λ ) ≤ − C | σ || λ | / ) | λ |≤ − ( (cid:37) − / C ) | σ | , λ ∈ I . (9.116)For the second inequality we have used (9.112a), while for the last inequality wehave estimated λ ≤ λ = 2 − (cid:37) | σ | . Taking (cid:37) sufficiently large yields Λ( I ) ⊂ (cid:98) C .In order to show Λ( I ) ⊂ (cid:98) C + we split I = [ λ , λ ] into two parts, [ λ , and (4 , λ ] (note that [ λ , ⊂ I by (9.107)). In the first part we estimate similarlyas in (9.116) to get Re Λ( λ ) ≥ − − C | σ | λ / ) λ ≥ (cid:37) − / C ) | σ | , λ ≤ λ ≤ . (9.117)Taking (cid:37) ∼ large enough the right most expression is larger than . If λ > ,we use the rough bound (9.112c) similarly as in (9.115) to obtain Re Λ( λ ) = − − (1 + µ ( λ )) λ ≥ − λ > , < λ ≤ λ . Together with (9.117) this shows that Λ( I ) ⊂ (cid:98) C + .Next, we will determine the three values a k using the four selection principlesof Lemma 9.9. Choice of a : The initial condition, i.e., SP-2 , must be satisfied, (cid:98) Ω a ( − −
0) = (cid:98) Ω a (Λ(0 − − . This excludes the choice a = 0 since (cid:98) Ω ( − −
0) = 2 . The choice a = − isexcluded using / -Hölder continuity (9.110) of the roots (9.101a) inside thedomain (cid:98) C − , and (9.112b): Im (cid:98) Ω − (Λ( − ξ )) = Im (cid:104) (cid:98) Ω − ( − − ξ ) + O (cid:0) | µ ( − ξ ) ξ | / (cid:1)(cid:105) ≤ − c ξ / , ≤ ξ ≤ . (9.118) 103or the last bound we have used (9.112a) and the bound ± Im (cid:98) Ω ± ( 1 + ξ ) = ± Im (cid:98) Ω ± ( − − ξ ) ≥ c ξ / , ≤ ξ ≤ , (9.119)which follows from the explicit formulas (9.101a). Since (9.118) violates SP-3 we are left with only one choice: a = + . Choice of a : Since (cid:98) Ω + ( − , while Ω(0) = − , we exclude the choice a = + using SP-2 . Moreover, from the explicit formulas of the roots (9.101a) itis easy to see that Im (cid:98) Ω a | ( − , = 0 for each of the three roots a = ± , . Similarlyas in (9.118) we estimate for small enough λ > the real and imaginary part of (cid:98) Ω ◦ Λ by Re (cid:98) Ω (Λ( λ )) ≤ − − c λ / + C | σ | / λ / (cid:12)(cid:12) Im (cid:98) Ω (Λ( λ )) (cid:12)(cid:12) = (cid:12)(cid:12) O (cid:0) | µ ( λ ) λ | / (cid:1)(cid:12)(cid:12) (cid:46) | σ | / λ / . (9.120)If a = 0 , then (9.120) would violate SP-4 for small λ > . We are left withonly one choice: a = − . Choice of a : Using the formulas (9.101a) we get (cid:8) (cid:98) Ω (1 ± , (cid:98) Ω + (1 ± , (cid:98) Ω − (1 ± (cid:9) = { , − } . Thus, the / -Hölder regularity (9.110) of the roots (outside the branch cuts)implies dist (cid:0) (cid:98) Ω a ( ζ ) , { , − } (cid:1) (cid:46) | ζ − | / , ζ ∈ C , a = 0 , ± . (9.121)We will apply this estimate for ζ = Λ( λ ) = 1 + O (cid:0) | λ − | + | σ | (cid:1) , λ ∈ [ λ , λ ] . Using (9.112a) to estimate µ ( λ ) , and recalling that | λ − | (cid:46) | σ | , for λ ∈ [ λ , λ ] ,(9.103) and (9.121) yield dist (cid:0) Ω( λ ) , { , − } (cid:1) ≤ max a dist (cid:0) (cid:98) Ω a (Λ( λ )) , { , − } (cid:1) (cid:46) | σ | / , λ ∈ [ λ , λ ] . (9.122)In particular, taking σ ∗ ∼ sufficiently small (9.122) implies for every | σ | ≤ σ ∗ , Ω([ λ , λ ]) ⊂ B (1 , ∪ B ( − , , where B ( ζ, ρ ) ⊂ C is a complex ball of radius ρ centered at ζ . Since a = − and (cid:98) Ω − (1 −
0) = 1 we see that Ω( λ − ∈ B (1 , . The continuity of Ω (cf. SP-1 )thus implies
Ω([ λ , λ ]) ⊂ B (1 , . In particular, | Ω( λ ) − | ≤ , while | (cid:98) Ω (Λ( λ )) − | ≥ , since (cid:98) Ω (1 + 0) = 2 and Λ( λ ) ∈ (cid:98) C + is close to . This shows that a (cid:54) = 0 .104n order to choose a among ± we use (9.110) and the symmetry Im (cid:98) Ω − = − Im (cid:98) Ω + to get ± Im (cid:98) Ω ± (Λ( λ )) ≥ Im (cid:98) Ω + ( − λ ) − C | λ µ ( λ ) | / , λ ∈ I . (9.123)Since λ = 2 + (cid:37) | σ | ≤ combining (9.119) and (9.112a) yields ± Im (cid:98) Ω ± (Λ( λ )) ≥ c ( λ − / − C | σ | / = ( c (cid:37) / − C ) | σ | / . (9.124)Taking (cid:37) ∼ sufficiently large, the last lower bound becomes positive. Thus,the choice: a = − is excluded by SP-3 . We are left with only one choice a = + . The estimate (9.109) follows from (9.124).For the rest of the analysis we always assume that the triple ( δ, (cid:37), σ ∗ ) isfrom Lemma 9.14. Next we determine the shape of the general edge when theassociated gap in supp v is small. Lemma 9.16 (Edge shape) . Let τ ∈ ∂ supp v and suppose | σ ( τ ) | ≤ σ ∗ , where σ ∗ ∼ is from Lemma 9.14. Then σ = σ ( τ ) (cid:54) = 0 , and supp v continues in thedirection sign σ such that (cid:12)(cid:12) Ω( λ ) − (cid:98) Ω + (1 + | λ | ) (cid:12)(cid:12) (cid:46) | σ | min (cid:8) | λ | , | λ | / (cid:9) , sign λ = sign σ . (9.125) In particular,
Im Ω( λ ) = Ψ edge (cid:16) | λ | (cid:17) + O (cid:16) | σ | min (cid:8) | λ | , | λ | / (cid:9)(cid:17) , sign λ = sign σ , (9.126) where the function Ψ edge : [0 , ∞ ) → [0 , ∞ ) , defined in (2.14a) , satisfies Ψ edge ( λ ) = Im (cid:98) Ω + (1 + 2 λ ) , λ ≥ . (9.127)We remark that from (2.14a) one obtains: Ψ edge ( λ ) ∼ min (cid:8) λ / , λ / (cid:9) , λ ≥ . (9.128) Proof of Lemma 9.16.
The bound σ (cid:54) = 0 follows from Lemma 9.12. Thestatement concerning the direction of supp v follows from Lemma 9.11. Withoutloss of generality we assume σ > . Let δ, σ ∗ ∼ be from Lemma 9.14. Therelation (9.125) is trivial when | λ | (cid:38) δ/ | σ | since Ω( λ ) and (cid:98) Ω + (1 + λ ) are both O ( λ / ) by (9.13a) and (9.101), respectively. Thus, we consider only the case λ ∈ I = (0 , λ ] . Using (9.108) and the stability estimate (9.111), with ρ = 1 ,we get Ω( λ ) = (cid:98) Ω + (1 + λ + µ ( λ ) λ )= (cid:98) Ω + (1 + λ ) + O (cid:16) µ ( λ ) min (cid:8) λ / , λ / (cid:9)(cid:17) , λ ∈ I = ( 0 , λ ] . (9.129) 105rom (9.112d) we obtain | µ ( λ ) | (cid:46) | σ | (cid:12)(cid:12) (cid:98) Ω + (1 + (1 + µ ( λ )) λ ) − (cid:98) Ω + (1 + 0) (cid:12)(cid:12) + | σ | λ . (9.130)The stability estimate (9.110) then yields (cid:12)(cid:12) (cid:98) Ω + (1 + (1 + µ ( λ )) λ ) − (cid:98) Ω + (1 + 0) (cid:12)(cid:12) (cid:46) min (cid:110)(cid:12)(cid:12) (1 + µ ( λ )) λ (cid:12)(cid:12) / , (cid:12)(cid:12) (1 + µ ( λ )) λ (cid:12)(cid:12) / (cid:111) (cid:46) min (cid:8) λ / , λ / (cid:9) , (9.131)where we have used the first estimate of (9.109) to obtain | (1 + µ ( λ )) λ | ∼ λ .Plugging (9.131) into (9.130) and using the resulting bound in (9.129) to esti-mate µ ( λ ) yields (9.125). The formula (9.126) follows by taking the imaginarypart of (9.125) and using (9.127). In order to see that (9.127) is equivalentto our original definition (2.14a) of Ψ edge ( λ ) we rewrite the right hand side of(9.127) using (9.101a) and (9.101b).We know now already from Lemma 9.14 that Im Ω is small in I since a = − and Im (cid:98) Ω − ( − λ ) = 0 , λ ∈ I . The next result shows that actually Im Ω | I = 0 which bounds the size of the gap ∆( τ ) from below. Lemma 9.17 (Size of small gap) . Suppose τ ∈ ∂ supp v . Then the gap length ∆( τ ) (cf. (9.59) ) is approximated by (cid:98) ∆( τ ) for small | σ ( τ ) | , such that ∆( τ ) (cid:98) ∆( τ ) = 1 + O (cid:0) σ ( τ ) (cid:1) . (9.132) In general ∆( τ ) ∼ | σ ( τ ) | (cid:46) (cid:98) ∆( τ ) . Proof.
Let ( δ, (cid:37), σ ∗ ) be from Lemma 9.14. If σ = σ ( τ ) satisfies | σ | ≥ σ ∗ , then ∆ = ∆( τ ) (cid:38) | σ | by the second line of (9.72). On the other hand, ∆ ≤ and | σ | (cid:46) by definitions (9.59) and (8.11), respectively. Thus, we find ∆ ∼ | σ | .Since ψ = ψ ( τ ) (cid:46) we see from (9.94) that (cid:98) ∆ = (cid:98) ∆( τ ) (cid:38) | σ | . Thus, thelemma holds for | σ | ≥ σ ∗ . Therefore from now on we will assume < | σ | ≤ σ ∗ ( σ (cid:54) = 0 by Lemma 9.16). Moreover, it suffices to consider only the case σ < without loss of generality.Let us define the gap length λ = λ ( τ ) in the normal coordinates as λ := inf (cid:8) λ > λ ) > (cid:9) . (9.133)Comparing this with (9.59) shows λ = 2 ∆ (cid:98) ∆ . (9.134)From (9.109) we already see that λ ≤ λ , which is equivalent to ∆ ≤ (1 + (cid:37) | σ | ) (cid:98) ∆ . (9.135) 106ince (cid:37) ∼ the estimate (9.132) hence follows if we prove the lower bound, ∆ ≥ (1 − C | σ | ) (cid:98) ∆ . (9.136)Using the representation (9.103) and the perturbation bound (9.110) we get Im Ω( λ ) = Im (cid:98) Ω − ( − λ ) + O (cid:0) | λµ ( λ ) | / (cid:1) ≤ C | σ | / , ∀ λ ∈ I . (9.137)We will show that λ (cid:55)→ Im Ω( λ ) , grows at least like a square root function onthe domain { λ : Im Ω( λ ) ≤ cε } . More precisely, we will show that if λ ≤ ,then Im Ω( λ + ξ ) (cid:38) ξ / , ≤ ξ ≤ . (9.138)Assuming that (9.138) is known, the estimate (9.136) follows from (9.137) and(9.138). Indeed, if λ ≥ λ = 2 − (cid:37) | σ | then (9.137) is immediate as (cid:37) ∼ . Onthe other hand, if λ < λ , then c ( λ − λ ) / ≤ Im Ω( λ ) ≤ C | σ | / . Solving this for λ yields λ ≥ λ − ( C /c ) | σ | ≥ − C | σ | , where λ = 2 − (cid:37) | σ | with (cid:37) ∼ (cf. (9.105)) has been used to get the lastestimate. Using (9.134) we see that this equals (9.136). Together with (9.135)this proves (9.132).In order to prove the growth estimate (9.138), we express it in the originalcoordinates ( ω, v ( τ + ω )) using (9.96), (9.8), v ( τ + ∆) = 0 , and f, | m | ∼ (note that b = f since v ( τ ) = 0 ): v ( τ + ∆ + (cid:101) ω ) (cid:38) min (cid:110)(cid:0) (cid:98) ∆( τ ) − / (cid:1) (cid:101) ω / , (cid:101) ω / (cid:111) , ≤ (cid:101) ω ≤ δ . (9.139)Applying Lemma 9.16 with τ + ∆ as the base point yields v ( τ + ∆ + (cid:101) ω ) ∼ min (cid:110)(cid:0) (cid:98) ∆( τ + ∆ ) − / (cid:1) (cid:101) ω / , (cid:101) ω / (cid:111) , ≤ (cid:101) ω ≤ δ . (9.140)The relation (9.140) implies (9.139), provided we show (cid:98) ∆( τ + ∆) (cid:46) (cid:98) ∆( τ ) , for ∆ (cid:46) (cid:98) ∆( τ ) . (9.141)From the definition (9.94) we get (cid:98) ∆( τ + ∆) ∼ | σ ( τ + ∆) | ψ ( τ + ∆) . (9.142) 107sing the upper bound (9.135) and (9.95) we see that ∆ (cid:46) (cid:98) ∆( τ ) ∼ | σ ( τ ) | , for sufficiently small σ ∗ ∼ . Since σ ( τ ) is / -Hölder continuous in τ , we get | σ ( τ + ∆) | ≤ | σ ( τ ) | + C ∆ / (cid:46) | σ ( τ ) | . (9.143)From the stability of the cubic (9.12) it follows that for small enough σ ∗ ∼ we have ψ ( τ + ∆) ∼ ψ ( τ ) ∼ . Plugging this together with (9.143) into (9.142) yields (9.141).We have now covered all the parameter regimes of σ and ψ satisfying (9.12).Combining the preceding lemmas yields the expansion around general basepoints τ where v ( τ ) = 0 . We will need the following representation of theedge shape function (2.14a) below: Ψ edge ( λ ) = λ / √ (cid:101) Ψ( λ )) , λ ≥ , (9.144)where the smooth function (cid:101) Ψ : [0 , ∞ ) → R has uniformly bounded derivatives,and (cid:101) Ψ(0) = 0 . Proof of Proposition 9.8.
Let τ ∈ supp v satisfy v ( τ ) = 0 . If σ ( τ ) = 0 ,then the expansion (9.62) follows directly from Lemma 9.12.In the case < | σ ( τ ) | ≤ σ ∗ (9.126) in Lemma 9.16 yields (9.60) with (cid:98) ∆ = (cid:98) ∆( τ ) in place of ∆ = ∆( τ ) . Here, the threshold σ ∗ ∼ is fixed byLemma 9.14. We will show that replacing (cid:98) ∆ with ∆ in (9.60) yields an errorthat is so small that it can be absorbed into the sub-leading order correction of(9.60). Since the smooth auxiliary function (cid:101) Ψ in the representation (9.144) of Ψ edge has uniformly bounded derivatives, we get for every ≤ λ (cid:46) , Ψ edge ((1 + (cid:15) ) λ ) = (1 + (cid:15) ) / Ψ edge ( λ ) + O (cid:0) (cid:15) min (cid:8) λ / , λ / (cid:9)(cid:1) , λ ≥ , (9.145)provided the size | (cid:15) | (cid:46) of (cid:15) ∈ R is sufficiently small. On the other hand, if | λ | (cid:38) then (9.145) follows from (9.111) of Lemma 9.15. Now by Lemma 9.17 wehave (cid:98) ∆ = (1 + | σ | κ ) ∆ , where ∆ = ∆( τ ) and the constant κ ∈ R is independentof λ , and can be assumed to satisfy | κ | ≤ / (otherwise we reduce σ ∗ ∼ ).Thus applying (9.145) with (cid:15) = | σ | κ = O ( ∆ / ) , yields | σ | Ψ edge (cid:18) ω (cid:98) ∆ (cid:19) = ( 1 + | σ | κ ) / | σ | ∆ / ∆ / Ψ edge (cid:18) ω ∆ (cid:19) + O (cid:18) min (cid:26) | ω | / ∆ / , | ω | / (cid:27)(cid:19) , for ω ≥ . Here, the error on the right hand side is of smaller size than thesubleading order term in the expansion (9.60).108rom (9.15) we identify the formula for h x , in the case < | σ | ≤ σ ∗ : h x := (1+ | σ | κ ) / ψ | σ | ∆ / | m x | f x when < | σ | ≤ σ ∗ ; (cid:113) / | σ | h (cid:48) x when | σ | > σ ∗ . (9.146)For | σ | ≤ σ ∗ we used (9.126). In the case | σ | > σ ∗ , the function h (cid:48) x is from(9.72), and the function h is defined such that h (cid:48) x (cid:12)(cid:12)(cid:12) ωσ (cid:12)(cid:12)(cid:12) / = h x ∆ / Ψ edge (cid:18) ω ∆ (cid:19) + O (cid:18) | ω | / ∆ / (cid:19) . (9.147)Here, the second term originates from the representation (9.144) of Ψ edge . Thisproves (9.60).Finally, suppose τ and τ are the opposite edges of supp v , separated by asmall gap of length ∆ (cid:46) σ ∗ , between them. Now, f ( τ ) , | m ( τ ) | and ψ ( τ ) are / -Hölder continuous in τ , and satisfy f, | m | , ψ ∼ . Thus, the terms constituting h x in the case | σ | ≤ σ ∗ in (9.146) satisfy f x ( τ ) f x ( τ ) = 1 + O ( ∆ / ) , | m x ( τ ) || m x ( τ ) | = 1 + O ( ∆ / ) , ψ ( τ ) ψ ( τ ) = 1 + O ( ∆ / ) . (9.148)Of course, ∆ = ∆( τ ) = ∆( τ ) . Moreover, by Lemma 9.17, (cid:98) ∆( τ ) (cid:98) ∆( τ ) = 1 + O (∆ / ) . (9.149)Using (9.94) we express | σ | in terms of (cid:98) ∆ , f, | m | , ψ , and hence (9.148) and (9.149)imply | σ ( τ ) || σ ( τ ) | = 1 + O ( ∆ / ) . (9.150)Thus, combining (9.148), (9.149), and (9.150), we see from (9.146) that h ( τ ) = h ( τ ) + O B ( ∆ / ) . This proves the last remaining claim of the proposition. Pick ε > , and recall the definitions (9.1) and (9.3) of D ε and M ε , respectively.In the following we split M ε into two parts: M (1) := ∂ supp v M (2) ε := M ε \ ∂ supp v . (9.151) Proof of Theorem 2.6.
Combining Proposition 9.3 and Proposition 9.8 showsthat there are constants ε ∗ , δ , δ ∼ such that the following hold:109. If τ ∈ M (1) , then σ ( τ ) (cid:54) = 0 and v x ( τ + ω ) ≥ c | ω | / , for ≤ sign σ ( τ ) ω ≤ δ .2. If τ ∈ M (2) ε ∗ , then v x ( τ + ω ) ≥ c (cid:0) v x ( τ ) + | ω | / (cid:1) , for − δ ≤ ω ≤ δ .In the case 1 each connected component of supp v must be at least of length δ ∼ . This implies (2.15). In particular, by combining (2.7) and (5.8) we seethat supp v is contained in an interval of length , and therefore the numberof the connected components K (cid:48) satisfies K (cid:48) ∼ .In order to prove (2.18) and (2.19) we may assume that ε ≤ ε ∗ and | ω | ≤ δ for some ε ∗ , δ ∼ . Indeed, (2.18) becomes trivial when Cε ≥ . Similarly, if (cid:104) v ( τ ) (cid:105) + | ω | (cid:38) , then (cid:104) v ( τ ) (cid:105) + Ψ( ω ) ∼ and thus the O ( · · · ) -term in (2.19)is O (1) . Since v ≤ ||| m ||| R ∼ , the expansion (2.19) is hence trivial.Obviously the bounds in the cases 1. and 2. continue to hold if we reducethe parameters ε ∗ , δ , δ . We choose ε ∗ ∼ so small that ( ε ∗ /c ) ≤ δ and ( ε ∗ /c ) ≤ δ . Let us define the expansion radius around τ ∈ M ε for every ε ≤ ε ∗ δ ε ( τ ) := (cid:40) ( ε/c ) if τ ∈ M (1) ( ε/c ) if τ ∈ M (2) ε , (9.152)and the corresponding expansion domains I ε ( τ ) := (cid:40)(cid:8) τ + sign σ ( τ ) ξ : 0 ≤ ξ ≤ δ ε ( τ ) (cid:9) if τ ∈ M (1) (cid:2) τ − δ ε ( τ ) , τ + δ ε ( τ ) (cid:3) if τ ∈ M (2) ε . (9.153)If τ ∈ I ε ( τ ) for some τ ∈ M ε then either v x ( τ ) ≥ c | τ − τ | / or v x ( τ ) ≥ c | τ − τ | / depending on whether τ is an edge or not. In particular, it followsthat (cid:104) v ( τ ) (cid:105) ≥ ε , ∀ τ ∈ ∂I ε ( τ ) \ ∂ supp v . (9.154)This implies that each connected component of D ε is contained in the expansiondomain I ε ( τ ) of some τ ∈ M ε , i.e., D ε ⊂ (cid:91) τ ∈ M ε I ε ( τ ) . (9.155)In order to see this formally let τ ∈ D ε \ M ε be arbitrary, and define τ ∈ M ε asthe nearest point of M ε from τ , in the direction, θ := − sign ∂ τ (cid:104) v ( τ ) (cid:105) , where (cid:104) v (cid:105) decreases. In other words, we set τ := τ + θ ξ , where ξ := inf (cid:8) ξ > τ + θ ξ ∈ M ε (cid:9) . (9.156) 110rom (9.156) it follows that if τ ∈ ∂ supp v , then supp v continues in the direc-tion sign( τ − τ ) = − θ from τ . We show that | τ − τ | ≤ δ ε ( τ ) . To this end,suppose | τ − τ | > δ ε ( τ ) , and define τ := τ + sign( τ − τ ) δ ε ( τ ) , (9.157)as the point between τ and τ exactly at the distance δ ε ( τ ) away from τ . Now, τ / ∈ ∂ supp v as otherwise τ would not be the nearest point of M ε (cf. (9.156)).On the other hand, by definition we have τ ∈ ∂I ( τ ) . Thus, the estimate(9.154) with τ in place of τ yields (cid:104) v ( τ ) (cid:105) ≥ ε ≥ (cid:104) v ( τ ) (cid:105) . Since (cid:104) v (cid:105) is continuously differentiable on the set where (cid:104) v (cid:105) > and ( τ − τ ) ∂ τ (cid:104) v ( τ ) (cid:105) < by (9.156) and (9.157), we conclude that (cid:104) v (cid:105) has a local minimumat some point τ ∈ M ε lying between τ and τ . But this contradicts (9.156). As τ ∈ D ε \ M ε was arbitrary (9.155) follows.From Corollary 9.4 we know that for every τ , τ ∈ M (2) ε , either | τ − τ | ≥ c or | τ − τ | ≤ C ε , (9.158)holds. Let { γ k } be a maximal subset of M (2) ε such that its elements are separatedat least by a distance c . Then the set M := ∂ supp v ∪ { γ k } has the propertiesstated in the theorem. In particular, D ε ⊂ (cid:91) τ ∈ ∂ supp v I ε ( τ ) ∪ (cid:91) k (cid:2) γ k − Cε , γ k + Cε (cid:3) , since M (2) ε + [ − Cε , Cε ] ⊂ ∪ k [ γ k − Cε , γ k + 2 Cε ] for sufficiently small ε ∼ . This completes the proof of Theorem 2.6.Next we show that the support of a bounded generating density is a singleinterval provided the rows of S can not be split into two well separated subsets.We measure this separation using the following quantity ξ S ( κ ) := sup (cid:40) inf x ∈ Ay / ∈ A (cid:107) S x − S y (cid:107) : κ ≤ π ( A ) ≤ − κ, A ⊂ X (cid:41) , κ ≥ . (9.159) Lemma 9.18 (Generating density supported on single interval) . Assume S satisfies A1-3 and ||| m ||| R ≤ Φ for some Φ < ∞ . Considering Φ as an additionalmodel parameter, there exist ξ ∗ , κ ∗ ∼ , such that under the assumption, ξ S ( κ ∗ ) ≤ ξ ∗ , (9.160) the conclusions of Theorem 2.11 hold. S which do not satisfy(9.160) and the associated generating density v is shown to have a non-connectedsupport. Proof of Theorem 2.11.
Let ξ ∗ , κ ∗ ∼ be from Lemma 9.18. Note that(2.27) is equivalent to ξ S (0) ≤ ξ ∗ , and ξ S ( κ (cid:48) ) ≤ ξ S ( κ ) , whenever κ (cid:48) > κ . Thus(2.27) implies ξ S ( κ ∗ ) ≤ ξ S (0) ≤ ξ ∗ , and hence the theorem follows from thelemma. Proof of Lemma 9.18.
Since ||| m ||| R ≤ Φ Theorem 2.6 yields the expansion(2.28b) around the extreme edges α := inf supp v and β := sup supp v . Inparticular, there exists δ ∼ such that v x ( α + ω ) ≥ c | ω | / and v x ( β − ω ) ≥ c | ω | / , for ω ∈ [0 , δ ] . (9.161)Let us write m x ( τ ) = p x ( τ ) u x ( τ ) + i v x ( τ ) where p x = sign Re m x ∈ {− , +1 } and u x := | Re m x | , v x = Im m x ≥ . Bycombining the uniform bound ||| m ||| R ≤ Φ with (5.9) we see that | m x | ∼ . Inparticular, there exists ε ∗ ∼ such that max { u x , v x } ≥ ε ∗ . (9.162)Since m x ( τ ) is continuous in τ , the constraint (9.162) means that Re m x ( τ ) cannot be zero on the domain K := (cid:110) τ ∈ [ − Σ , Σ ] : sup x v x ( τ ) ≤ ε ∗ (cid:111) . If I is a connected component of K , then there is p Ix ∈ {− , +1 } , x ∈ X , suchthat p ( τ ) = p I , ∀ τ ∈ I .
Using (9.161) we choose ε ∗ ∼ to be so small that v x ( α + δ ) and v x ( α − δ ) areboth larger than ε ∗ . It follows that supp v is not contained in K . Furthermore,we choose ε ∗ so small that Lemma 9.2 applies, i.e., v x > grows monotonicallyin K when Π ≥ Π ∗ .We will prove the lemma by showing that if some connected component I of K satisfies, I = [ τ , τ ] ⊂ K , where α + δ ≤ τ < τ ≤ β − δ , (9.163)then the set A = A I := (cid:8) x ∈ X : p Ix = +1 (cid:9) (9.164) 112atisfies π ( A ) ∼ (9.165a) (cid:107) S x − S y (cid:107) ∼ , x ∈ A , y / ∈ A . (9.165b)The estimates (9.165) imply ξ S ( κ ∗ ) ≥ ξ ∗ , with κ ∗ = π ( A ) and ξ ∗ ∼ . In otherwords, under the assumption (9.160) each connected component of K containseither α or β . Together with (2.28b) this proves the remaining estimate (2.28a)of the lemma, and the supp v is a single interval.In order to prove (9.165a) we will show below that there is a point τ ∈ I such that | σ ( τ ) | ≤ C ε ∗ , (9.166)where σ := (cid:104) pf (cid:105) was defined in (8.11). Let f − := inf x f x and f + := sup x f x .As m is uniformly bounded, Proposition 5.3 shows that f ± ∼ . Hence, (9.166)yields bounds on the size of A , π ( A ) f − (1 − π ( A )) f − ≥ σ ( τ ) ≥ − C ε ∗ π ( A ) f − − (1 − π ( A )) f ≤ σ ( τ ) ≤ + C ε ∗ . Solving for π ( A ) , we obtain f − − C ε ∗ f + f − ≤ π ( A ) ≤ f + C ε ∗ f + f − . By making ε ∗ ∼ sufficiently small this yields (9.165a).We now show that there exists τ ∈ I satisfying (9.166). To this end weremark that at least one (actually exactly one) of the following three alternativesholds true:(a) The interval I contains a non-zero local minimum τ of (cid:104) v (cid:105) .(b) The interval I contains a left and right edge τ − ∈ ∂ supp v and τ + ∈ ∂ supp v .(c) The average generating density (cid:104) v (cid:105) has a cusp at τ ∈ I ∩ (supp v \ ∂ supp v ) such that v ( τ ) = σ ( τ ) = 0 .In the case (a), since m is smooth on the set where (cid:104) v (cid:105) > , Lemma 9.2 implies Π( τ ) < Π ∗ , and thus (9.166) holds for C ≥ Π ∗ . In the case (b) we know that ± σ ( τ ± ) > by Proposition 9.8. Since σ ( τ ) is continuous (cf. Lemma 9.1), therehence exists τ ∈ ( τ − , τ + ) ⊂ I such that σ ( τ ) = 0 . Finally, in the case (c) wehave σ ( τ ) = 0 by Proposition 9.8.Now we prove (9.165b). Since v x ≤ u x ≤ | m x | ≤ Φ on I , and m solves theQVE, we obtain for every x ∈ A , y / ∈ A and τ ∈ I ≤ u x + 1 u y ≤ u x + u y | m x m y | ≤ | ( u x + u y ) + i ( v x − v y ) || m x || m y | = 2 (cid:12)(cid:12)(cid:12) m x − m y (cid:12)(cid:12)(cid:12) = 2 |(cid:104) S x − S y , m (cid:105)| ≤ (cid:107) S x − S y (cid:107) . (9.167) 113ere, the definition (9.164) of A is used in the first bound while u x ≥ v x wasused in the second estimate. The bound (9.167) is equivalent to (9.165b) as (cid:107) S x − S y (cid:107) ≥ / (2Φ ) ∼ .We have shown that | σ | + (cid:104) v (cid:105) ∼ . By using this in Corollary 7.3 we see that v ( τ ) is uniformly / -Hölder continuous everywhere.114 hapter 10 Stability around smallminima of generating density
The next result will imply the statement (ii) in Theorem 2.12. Since it plays acentral role in the proof of local laws (cf. Chapter 3) for random matrices in[AEK16c], we state it here in the form that does not require any knowledge ofthe preceding expansions and the associated cubic analysis. In fact, togetherwith our main results, Theorem 2.4 and Theorem 2.6, the next proposition isthe only information we use in [AEK16c] concerning the stability of the QVE.
Proposition 10.1 (Cubic perturbation bound around critical points) . Assume S satisfies A1-3 , ||| m ||| R ≤ Φ , for some Φ < ∞ , and g, d ∈ B satisfy theperturbed QVE (2.29) at some fixed z ∈ H . There exists ε ∗ ∼ such that if (cid:104) Im m ( z ) (cid:105) ≤ ε ∗ , and (cid:107) g − m ( z ) (cid:107) ≤ ε ∗ , (10.1) then there is a function s : H → B depending only on S and a , and satisfying (cid:107) s ( z ) (cid:107) (cid:46) , (cid:107) s ( z ) − s ( z ) (cid:107) (cid:46) | z − z | / , ∀ z , z ∈ H , (10.2) such that the modulus of the complex variable Θ = (cid:10) s ( z ) , g − m ( z ) (cid:11) (10.3) bounds the difference g − m ( z ) , in the following senses: (cid:107) g − m ( z ) (cid:107) (cid:46) | Θ | + (cid:107) d (cid:107) (10.4a) |(cid:104) w, g − m ( z ) (cid:105)| (cid:46) (cid:107) w (cid:107) | Θ | + (cid:107) w (cid:107)(cid:107) d (cid:107) + |(cid:104) T ( z ) w, d (cid:105)| , ∀ w ∈ B . (10.4b) Here the linear operator T ( z ) : B → B depends only on S and a , in additionto z , and satisfies (cid:107) T ( z ) (cid:107) (cid:46) . Moreover, Θ satisfies a cubic inequality (cid:12)(cid:12) | Θ | + π | Θ | + π | Θ | (cid:12)(cid:12) (cid:46) (cid:107) d (cid:107) + |(cid:104) t (1) ( z ) , d (cid:105)| + |(cid:104) t (2) ( z ) , d (cid:105)| , (10.5) 115 here t ( k ) : H → B , k = 1 , , depend on S , a , and z only, and satisfy (cid:107) t ( k ) ( z ) (cid:107) (cid:46) . The coefficients, π and π , may depend on S , z , a , as well as on g . Theysatisfy the estimates, | π | ∼ (cid:104) Im m ( z ) (cid:105) + | σ ( z ) | (cid:104) Im m ( z ) (cid:105) + Im z (cid:104) Im m ( z ) (cid:105) (10.6a) | π | ∼ (cid:104) Im m ( z ) (cid:105) + | σ ( z ) | , (10.6b) where the / -Hölder continuous function σ : H → [0 , ∞ ) is determined by S and a , and has the following properties: Let M = { α i } ∪ { β j } ∪ { γ k } be the set (2.17) of minima from Theorem 2.6, and suppose τ ∈ M satisfies | z − τ | = dist( z, M ) .If τ ∈ ∂ supp v = { α i } ∪ { β j } , then | σ ( α i ) | ∼ | σ ( β i − ) | ∼ ( α i − β i − ) / (10.7a) with the convention β = α − and α K (cid:48) +1 = β K (cid:48) + 1 . If τ / ∈ ∂ supp v = { γ k } ,then | σ ( γ k ) | (cid:46) (cid:104) Im m ( γ k ) (cid:105) . (10.7b) All the comparison relations depend only on the model parameters ρ , L , (cid:107) a (cid:107) , (cid:107) S (cid:107) L → B and Φ . We remark here that the coefficients π k do depend on g in addition to S and a , in contrast to the coefficients µ k in Proposition 8.2. The important point isthat the right hands sides of the comparison relations (10.6a) and (10.6b) arestill independent of g . This result is geared towards problems where d and g are random . Such problems arise when the resolvent method, as described inChapter 3, is used to study the local spectral statistics of Wigner-type randommatrices. The continuity and size estimates (10.2), (10.7a) and (10.7b) willbe used to extend high probability bounds for each individual z to all z in acompact set of H similarly as in the proof of Theorem 3.2. The various auxiliaryquantities, such as s, π ( k ) , T , etc., appearing in the proposition will be explicitlygiven in the proof, but their specific form is irrelevant for the applications, andhence we omitted them in the statement. Proof of Proposition 10.1.
Since z is fixed we write m = m ( z ) , etc. Bychoosing ε ∗ ∼ small enough we ensure that both Lemma 8.1 and Proposi-tion 8.2 are applicable. We choose s such that Θ becomes the component of u = ( g − m ) / | m | in the direction b exactly as in Proposition 8.2. Hence usingthe explicit formula (8.8) for the projector P we read off from Θ b = P u , that s := 1 (cid:104) b (cid:105) b | m | . (10.8)From Lemma 8.1 and Proposition 7.1 we see that this function has the properties(10.2). 116he first bound (10.4a) follows by using (8.29) and (8.30) in the definition(8.27a) of u . More precisely, we have (cid:107) g − m (cid:107) ≤ (cid:107) m (cid:107)(cid:107) u (cid:107) ≤ (cid:107) m (cid:107) (cid:0) | Θ |(cid:107) b (cid:107) + (cid:107) r (cid:107) (cid:1) (cid:46) | Θ | + (cid:107) d (cid:107) , where (cid:107) m (cid:107) ∼ , b = f + O B ( α ) , r = Rd + O B ( | Θ | + | d | ) , and (cid:107) R (cid:107) , (cid:107) f (cid:107) (cid:46) ,have been used.In order to derive (10.4b) we first write (cid:104) w, g − m (cid:105) = (cid:104)| m | w, u (cid:105) = (cid:104)| m | w, b (cid:105) Θ + (cid:104)| m | w, r (cid:105) . (10.9)Clearly, |(cid:104)| m | w, b (cid:105)| (cid:46) (cid:107) w (cid:107) . Moreover, using (8.29) we obtain (cid:104)| m | w, r (cid:105) = (cid:68) | m | w, Rd + O B ( | Θ | + (cid:107) d (cid:107) ) (cid:69) = (cid:104) R ∗ ( | m | w ) , d (cid:105) + O (cid:16) (cid:107) m (cid:107)(cid:107) w (cid:107) (cid:0) | Θ | + (cid:107) d (cid:107) (cid:1) (cid:17) . Plugging this into (10.9), and setting T := R ∗ ( | m | · ) , we recognize (10.4b). Thebound (8.30) yields (cid:107) T (cid:107) (cid:46) .As a next step we show that (10.5) and (10.6) constitute just a simplifiedversion of the cubic equation presented in Proposition 8.2. Combining (8.31)and (8.32) we get (cid:12)(cid:12) (cid:101) µ | Θ | + (cid:101) µ | Θ | + (cid:101) µ | Θ | (cid:12)(cid:12) (cid:46) |(cid:104)| m | b, d (cid:105)| + (cid:107) d (cid:107) + |(cid:104) e, d (cid:105)| , (10.10)where (cid:101) µ := (Θ / | Θ | ) µ , (cid:101) µ := (Θ / | Θ | ) µ and (cid:101) µ = (Θ / | Θ | ) µ + O ( | Θ | ) . Thelast term in the definition of (cid:101) µ accounts for the absorption of the O ( | Θ | ) -sizedpart of κ ( u, d ) in (8.31). Moreover, we have estimated the O ( | Θ | |(cid:104) e, d (cid:105)| ) -sizedpart of κ by a larger O ( |(cid:104) e, d (cid:105)| ) term. Recall that | Θ | (cid:46) ε ∗ from (10.1). Hencetaking ε ∗ ∼ small enough, the stability of the cubic (cf. (8.34)) implies thatthere is c ∼ so that | (cid:101) µ | + | (cid:101) µ | = | µ | + | µ | + O ( | Θ | ) ≥ c applies. Hencethe coefficients π := (cid:0) (cid:101) µ + ( (cid:101) µ − | Θ | (cid:1) (cid:8) | µ | ≥ c (cid:9) + (cid:101) µ (cid:101) µ (cid:8) | µ | < c (cid:9) π := (cid:101) µ (cid:8) | µ | ≥ c (cid:9) + (cid:101) µ (cid:101) µ (cid:8) | µ | < c (cid:9) , (10.11)scale just like µ and µ in size, i.e., | π | ∼ | µ | and | π | ∼ | µ | , provided ε ∗ and thus | Θ | is sufficiently small. Moreover, by construction the bound (10.10)is equivalent to (10.5) once we set t (1) := | m | ¯ b and t (2) := e .Let us first derive the scaling relation (10.6a) for π . Using σ ∈ R , we obtainfrom (8.33c): | π | ∼ | µ | = (cid:12)(cid:12)(cid:12) − (cid:104) f | m |(cid:105) ηα + i2 σ α − ψ − σ ) α + O (cid:0) α + η (cid:1)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12) (cid:104) f | m |(cid:105) ηα + ( ψ − σ ) α + O (cid:0) α + η (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) σα + O (cid:0) α + η (cid:1)(cid:12)(cid:12)(cid:12) . (10.12) 117e will now use the stability of the cubic, ψ + σ (cid:38) (cf. (8.34)). We treattwo regimes separately.First let us assume that σ ≤ ψ . In that case ψ ∼ , and we find | π | ∼ ηα + α + | σ | α + O (cid:0) α + η (cid:1) ∼ ηα + α + | σ | α . (10.13)In order to get the first comparison relation we have used the fact that ψ − σ ∼ ψ ∼ and (cid:104) f | m |(cid:105) ∼ and hence the first two terms on the right hand side ofthe last line in (10.12) can not cancel each other. The second comparison in(10.13) holds provided ε ∗ ∼ is sufficiently small, recalling α ∼ (cid:104) v (cid:105) ≤ ε ∗ (cf.(8.2), so that the error can be absorbed into the term η/α + α .Now we treat the situation when σ > ψ . In this case | σ | ∼ , and thus forsmall enough ε ∗ , we have | π | ∼ (cid:12)(cid:12)(cid:12) ηα + O ( α + η ) (cid:12)(cid:12)(cid:12) + α = ηα + α + O ( α + η ) ∼ ηα + α ∼ ηα + | σ | α + α . (10.14)Here, the first two terms in the last line of (10.12) may cancel each other butin that case both of the terms are O ( α ) and hence the size of | π | is given bythe term | σ | α ∼ α .The scaling behavior (10.6b) of π follows from (8.33b) using (cid:107) F (cid:107) L → L =1 − (cid:104) f | m |(cid:105) η/α ∼ (cf. (5.20) and (5.3)) and the stability of the cubic, | π | ∼ | µ | ∼ | σ | + | ψ − σ | α ∼ | σ | + α . (10.15)The formula (10.6a) now follows from (10.14) and (10.15) by using α ∼(cid:104) Im m (cid:105) . The quantity σ = σ ( z ) was proven to be / -Hölder continuous al-ready in Lemma 8.1. In order to obtain the relation (10.7a) we use (9.95) andLemma 9.17 to get | σ ( τ ) | ∼ (cid:98) ∆( τ ) / ∼ ∆( τ ) / , for τ ∈ ∂ supp v such that | σ ( τ ) | ≤ σ ∗ . On the other hand, if | σ ( τ ) | ≥ σ ∗ ,where the threshold parameter σ ∗ ∼ is from (9.95), then also ∆( τ ) ∼ . Thisproves (10.7a).In order to obtain (10.7b) we consider the cases v ( γ k ) = 0 and v ( γ k ) > separately. If v ( γ k ) = 0 then Lemma 9.12 shows that σ ( γ k ) = 0 . If v ( γ k ) > then ∂ τ (cid:104) v ( γ ) (cid:105)| τ = γ k = 0 . Lemma 9.2 thus yields | σ ( γ k ) | ≤ Π ∗ (cid:104) v ( γ k ) (cid:105) . Since Π ∗ ∼ this finishes the proof of (10.7b).Combining our two results concerning general perturbations, Lemma 5.11and Proposition 10.1, with scaling behavior of m ( z ) as described by Theo-rem 2.6, we now prove Theorem 2.12. Proof of Theorem 2.12.
Recall the definition (5.38) of operator B . We willshow below that (cid:107) B ( z ) − (cid:107) (cid:46) (cid:37) ( z ) + (cid:36) ( z ) / , | z | ≤ , (10.16) 118here (cid:37) = (cid:37) ( z ) and (cid:36) = (cid:36) ( z ) are defined in (2.33). Given (10.16) the assertion(i) of the theorem follows by applying Lemma 5.11 with Φ introduced in thetheorem and Ψ := ( (cid:37) + (cid:36) / ) − (cid:46) ε − , where the constant ε ∈ (0 , is from(2.30). If (cid:37) ≥ ε ∗ or (cid:36) ≥ ε ∗ for some ε ∗ ∼ , then (ii) follows similarly fromLemma 5.11 with Ψ ∼ . Therefore, in order to prove (ii) it suffices to assumethat (cid:37), (cid:36) ≤ ε ∗ for some sufficiently small threshold ε ∗ ∼ .We will take ε ∗ so small that Proposition 10.1 is applicable, and thus thecubic equation (10.5) can be written in the form (cid:12)(cid:12) | Θ | + π | Θ | + π | Θ | (cid:12)(cid:12) (cid:46) δ , (10.17)with δ = δ ( z, d ) ≤ (cid:107) d (cid:107) given in (2.33c) of Theorem 2.12. Combining the defi-nition (10.3) of Θ with the a priori bound (2.34) for the difference g − m , weobtain | Θ | ≤ (cid:107) s (cid:107)(cid:107) g − m (cid:107) (cid:46) λ ( (cid:36) / + ρ ) . (10.18)For the last step we used also (10.2). We will now show that if (10.18) holds forsufficiently small λ ∼ , then the linear term of the cubic (10.17) dominates inthe sense that | π | ≥ | π || Θ | , and | π | ≥ | Θ | . (10.19)Let us first establish (10.19) when τ = Re z ∈ supp v . From (10.18) and(10.6) we get | Θ | (cid:46) λ ( (cid:37) + η / ) (10.20) | π | (cid:38) ( | σ | + α ) α (10.21) | π | ∼ | σ | + α . (10.22)Here we have used the general property v x ∼ (cid:104) v (cid:105) ∼ α that always holds when ||| m ||| R (cid:46) Φ . Since τ ∈ supp v we have (cid:36) = η in (10.20). Let us show that (cid:37) + η / (cid:46) α . (10.23)To this end, let τ = τ ( z ) ∈ M ε ∗ be such that | τ − τ | = dist( τ, M ε ∗ ) (10.24)holds. If τ / ∈ ∂ supp v , then (d) of Corollary A.1 yields (10.23) immediately(take ω := τ − τ in the corollary). If on the other hand τ ∈ ∂ supp v , then (a)of Corollary A.1 yields (cid:37) + η / (cid:46) ω / ( ∆ + ω ) / + η / (cid:46) ( ω + η ) / ( ∆ + ω + η ) / ∼ α , where ∆ = ∆( τ ) is the gap length (9.59) associated to the point τ ∈ ∂ supp v satisfying (10.24). 119ombining (10.23) and (10.20) we get | Θ | (cid:46) λ α . Using this bound togetherwith (10.21) and (10.22) we obtain (10.19) for sufficiently small λ ∼ .Next we prove (10.19) when τ / ∈ supp v , i.e., (cid:37) = 0 . In this case (10.18) and(10.6) yield | Θ | (cid:46) λ (cid:36) / (10.25) | π | (cid:38) η/α (10.26) | π | (cid:46) . (10.27)By combining the parts (b) and (c) of Corollary A.1 we get α ∼ η (∆ + η ) / (cid:36) / (cid:46) η (cid:36) − / , (10.28)where ∆ = ∆( τ ) is the gap length (9.59) associated to the point τ ∈ ∂ supp v .For the last bound in (10.28) we used (cid:36) ∼ ω + η ≤ ∆ + η . Plugging (10.28)into (10.26) we get | π | (cid:38) (cid:36) / . (10.29)Using this together with (10.25) and (10.27) we obtain (10.19) also when τ / ∈ supp v .The estimates (10.19) imply | Θ | (cid:46) (cid:12)(cid:12) π Θ (cid:12)(cid:12) ∼ (cid:12)(cid:12) | Θ | + π | Θ | + π | Θ | (cid:12)(cid:12) . Using (10.17) we hence get | Θ | (cid:46) | π Θ | (cid:46) δ , from which it follows that | Θ | (cid:46) min (cid:26) δ | π | , δ / (cid:27) . (10.30)If τ / ∈ supp v we have (cid:37) = 0 and thus (10.29) can be written as | π | (cid:38) (cid:37) + (cid:36) / . (10.31)This estimate holds also when τ ∈ supp v . If the point τ = τ ( τ ) ∈ M ε ∗ satisfying (10.24) is not an edge of supp v , then (10.31) follows immediatelyfrom (d) of Corollary A.1 and from | π | (cid:38) α from (10.21). In order to get(10.31) when τ ∈ supp v and τ ∈ ∂ supp v we set ω = | τ − τ | and considerthe cases ω + η > c ∆ and ω + η ≤ c ∆ for some small c ∼ separately. If ω + η > c ∆ , then we get α ∼ ( ω + η ) / ∼ ω / + η / ∼ (cid:37) + η / , (10.32)using part (a) of Corollary A.1 in both the first and the last estimate. On theother hand, if ω + η ≤ c ∆ for sufficiently small c ∼ , then | σ | = | σ ( z ) | ≥ | σ ( τ ) | − C | τ − z | / (cid:38) ∆ / − C ( ω + η ) / ≥
12 ∆ / , (10.33) 120here we have used / -Hölder continuity of σ and the relation (10.7a) fromProposition 10.1. For the last bound we have used | τ − z | ∼ ω + η as well.Therefore, we have | σ | α ∼ ∆ / ( ω + η ) / (cid:38) ω / + η / (cid:38) (cid:37) + η / . (10.34)Here, we have used (a) of Corollary A.1 twice. Combining (10.32) and (10.34)we get | σ | α + α (cid:38) (cid:37) + (cid:36) / , τ ∈ supp v . (10.35)Using this in (10.21) yields (10.31) when τ ∈ ∂ supp v .By combining (10.30) and (10.31) we obtain | Θ | (cid:46) Υ , (10.36)with Υ = Υ( z, d ) defined in (2.36). The estimates (2.35) now follow from (10.4)using (10.36).We still need to prove (10.16). If τ ∈ supp v , then (5.40a) of Lemma 5.9shows (cid:107) B − (cid:107) (cid:46) | σ | + α ) α . Using (10.35) we get (10.16) when τ ∈ supp v . In the remaining case τ / ∈ supp v (10.16) reduces to (cid:107) B − (cid:107) (cid:46) (cid:36) − / . (10.37)In order to prove this we use (5.40a) to get the first bound below: (cid:107) B − (cid:107) ≤ (cid:107) B − (cid:107) L → L ≤ − (cid:107) F (cid:107) L → L (cid:46) αη . (10.38)For the second estimate we have used the definition (5.38) of B and the identity(5.20). Finally, for the third inequality we used (cid:104) f | m |(cid:105) ∼ to estimate −(cid:107) F (cid:107) L → L (cid:38) η/α . Using (10.28) in (10.38) yields (10.37). This completes theproof of (10.16). 121 hapter 11 Examples
In this chapter we present some simple examples that illustrate the need of var-ious assumption made on a and S . Recall that the assumptions A1-3 whereintroduced in the beginning of Chapter 2, and they are used extensively through-out this paper. Our main results are formulated under the additional assumptionthat m is bounded in B . Verifying this uniform boundedness was treated as aseparate problem in Chapter 6, and for this purpose the additional assumptions B1 and B2 along with the auxiliary function Γ were introduced. In particular,the non-effective uniform bounds of Theorem 2.10 were replaced by the cor-responding quantitative results in the form of Theorem 6.1 and Theorem 6.4,which rely on B1-2 and assumptions on Γ .In the following sections we will demonstrate how the properties A3 and B1 and the function Γ are used to effectively rule out certain ’bad’ behaviors of m ,by considering simple examples. Before going into details let us shortly commentthe remaining assumptions A1 , A2 and B2 , which we will not address anyfurther. The assumption A1 is structural in nature. It reflects the applicationswe have in mind, e.g., random matrix theory as explained in Chapter 3 andSection 3 of [AEK16b]. On the other hand, for a full analysis of Laplace-likeoperator on rooted trees (cf. Chapter 1) the assumption of symmetry of S shouldbe lifted. The smoothing assumption A2 was made for technical reasons. Itis appropriate for the random matrix theory as it generalizes the upper bound(3.1) appearing in the definition of Wigner-type random matrices. The property B2 on the other hand is a practical condition for easily obtaining an effective L -bound on the solution m when a (cid:54) = 0 (cf. Remark 6.5).Besides demonstrating how the solution m can become unbounded, and howto exclude such blow-ups with the right assumptions, we also provide three otherkinds of examples in this chapter. First, in Section 11.4 we show that althoughgenerally playing a secondary role to S in our analysis, the non-constant func-tion a can also affect the behavior of m significantly. Second, in Section 11.5we explain how to switch between different representations of a given QVE,and possibly reduce the dimensionality of the problem. Third, in Section 11.6we provide a very simple two parameter family of operators S , for which the122orresponding solution of the QVE with a = 0 , exhausts all the different localshapes of the generating density, described by our main result, Theorem 2.6.Most of the examples here are represented in the special setting where X isthe unit interval and π is the restriction of the Lebesgue measure to this interval,i.e., ( X , B , π ) := (cid:0) [0 , , B ([0 , , d x (cid:1) , (11.1)with B ([0 , denoting the standard Borel σ -algebra. Together, with the discretecase (3.2) this is the most common setup for the QVE. An example, where amore complicated setup is natural is [AZ08] (cf. also Subsection 3.4). The uniform primitivity assumption, A3 , was made to exclude choices of S that lead to an essentially decoupled system. Without sufficient coupling ofthe components m x in the QVE the components of the imaginary part of thesolution are not necessarily comparable in size, i.e., v x ∼ v y , may not hold (cf.(5.10) of Lemma 5.4). No universal growth behavior at the edge of the supportof the generating density, as described by Theorem 2.6, can be expected in thiscase, since the support of v x may not even be independent of x .The simplest such situation is if the components may be partitioned into twosubsets I and I c = X \ I , that are completely decoupled in the sense that S leavesinvariant the families of functions which are supported either on I or I c . In thiscase the QVE decouples into two independent QVEs. These independent QVEscan then be analyzed separately using the theory developed here. Assumption A3 also excludes a situation, where the functions supported on I are mappedto the function supported on the complement of I , and vice versa. This casehas an instability at the origin τ = 0 (cf. Lemma A.6 and Theorem A.4 in thediscrete setup) and requires a special treatment of the lowest lying eigenvalueof S (cf. [AEK]).Another example, illustrating why A3 is needed, is the case where a = 0 and the integral kernel of S is supported on a small band along the diagonal: S xy = ε − ξ ( x + y ) (cid:8) | x − y | ≤ ε/ (cid:9) . Here, ξ : R → (0 , ∞ ) is some smooth function and ε > is a constant. Forany fixed ε the operator S satisfies A1-3 and B1 . Also, the conditions B2 and Γ( ∞ ) = ∞ (cf. (6.3)) hold for the corresponding QVE. As ε approaches zero,however, the constant L from assumption A3 (among other model parameterssuch as (cid:107) S (cid:107) L → B from A2 ) diverge. In the limit, S becomes a multiplicationoperator and the QVE decouples completely, − m x ( z ) = z + ξ ( x ) m x ( z ) . m x ( z ) := ξ ( x ) − / m sc (cid:0) ξ ( x ) − / z (cid:1) , where m sc : H → H is the Stieltjes transform of Wigner’s semi-circle law (1.4).In particular, the support of the component v x of the generating density dependson x . B , outliers, and function Γ The purpose of this section is to illustrate the role of the auxiliary function Γ ,generated by the pair ( a, S ) through (6.2), in proving bounds for m in B . Wepresent two simple families of QVEs for which the solutions m are uniformlybounded in L , but for which the corresponding Γ ’s become increasingly inef-fective in converting these bounds into B -bounds for some members of thesefamilies. In both cases a few exceptional row functions, S x = ( y (cid:55)→ S xy ) , causedivergencies in the corresponding components, m x , of the solution. In the firstexample, the QVE can be solved explicitly and thus the divergence can be readoff from the solution formula. The second example is a bit more involved. Itillustrates a somewhat counterintuitive phenomenon of divergencies that mayarise if one smoothens out discontinuities of the integral kernel of S on smallscales. B : Let a = 0 . Consider the × - block constant integral operator S with thekernel S xy = λ { x ≤ δ, y > δ } + λ { y ≤ δ, x > δ } + { x > δ, y > δ } , (11.2)parametrized by two positive constants λ and δ . For any fixed values of λ > and δ ∗ ∈ (0 , / , the properties A1-3 and B1 hold uniformly for every δ ≤ δ ∗ .In particular, the solutions are uniformly bounded in L for δ ≤ δ ∗ , since thepart (i) of Theorem 6.1 yields a uniform bound when | z | ≤ ε , for some ε ∼ ,while (2.9) guarantees the L -boundedness in the remaining domain | z | > ε . Infact, the solution for any parameter values has the structure m x ( z ) = µ ( z ) { x ≤ δ } + ν ( z ) { x > δ } , (11.3)where the two functions µ, ν : H → H satisfy the coupled equations − µ ( z ) = z + (1 − δ ) λ ν ( z ) , − ν ( z ) = z + λ δ µ ( z ) + (1 − δ ) ν ( z ) . (11.4)Let us consider a fixed λ > . Then, as we take the limit δ ↓ the strictlyincreasing function Γ generated by S through (6.2), satisfies Γ( τ ) ≤ (cid:112) δτ , τ ∈ (0 , ∞ ) . (11.5) 124igure 11.1: As δ decreases the average generating density remains bounded,but the -th component of the generating density blows up at ± τ .This means that the uniform bound (6.6) becomes ineffective as Γ − (Λ) → ∞ for any fixed Λ ∈ (1 , ∞ ) as δ ↓ . Indeed, the row functions S x indexed by asmall set of rows x ∈ [0 , δ ] differ from the row functions indexed by x ≥ δ , andthis leads to a blow-up in the components m x ( z ) with x ∈ [0 , δ ] at a specificvalue of z . More precisely, we find | µ ( ± τ ) | ∼ √ δ , at τ := 2 λ (cid:112) λ − ( λ − . (11.6)While the B -norm of m diverges as δ approaches zero, the L -norm stays finite,because the divergent components contribute less and less. The situation isillustrated in Figure 11.1.The integral kernel (11.2) makes sense even for δ = 0 . In this case we getfor the generating measure the formulas, v (d τ ) = λ √ − τ λ − τ ( λ − { τ ∈ [ − , } d τ + π ( λ − λ − (cid:0) δ − τ (d τ ) + δ τ (d τ ) (cid:1) ,v x (d τ ) = 12 (cid:112) − τ { τ ∈ [ − , } d τ , x ∈ (0 , . The non-zero value that v assigns to τ and − τ reflects the divergence of m in the uniform norm at these points.In the context of random matrix theory the operator S with small valuesof the parameter δ corresponds to the variance matrix (cf. Definition 3.1) of aperturbation of a Wigner matrix. The part of the generating density, which issupported around τ corresponds to a small collection of eigenvalues away fromthe bulk of the spectrum of the random matrix. These outliers will induce adivergence in some elements of the resolvent (3.4) of this matrix. This divergenceis what we see as the divergence of µ in (11.6).125 B due to smoothing: We present a second example of a different nature, in which the bounds ofProposition 6.6 for converting L -estimates of m ( z ) into uniform bounds becomeineffective. The smoothing of discontinuities in S may cause blow-ups in thesolution of the QVE (cf. Figure 11.2). This is somewhat surprising, sinceby conventional wisdom, smoother data implies smoother solutions. The keypoint here is that the smoothing procedure creates a few row functions thatare far away from all the other row functions. The following choice of operatordemonstrates this mechanism: S ( ε ) xy = 12 ( r x s y + r y s x ) . Here the two continuous functions r, s : [0 , → (0 , , are given by r x = (cid:0) ε − ( x − δ ) (cid:1) { δ − ε < x ≤ δ } + { x > δ } ,s x = 2 λ { x ≤ δ } + (cid:0) λ − ε − (cid:0) λ − x − δ ) (cid:1) { δ < x ≤ δ + ε } + { x > δ + ε } , respectively. The parameters λ > , δ ∈ (0 , are considered fixed, while ε ∈ (0 , δ ) is varied. The continuous kernel S ( ε ) represents a smoothed outversion of the × -block operator S (0) = S from (11.2).In this case, Γ( ∞ ) = lim τ →∞ Γ( τ ) = ∞ holds for each operator S ( ε ) , ε > ,as well as for the limiting operator S (0) . However, the estimates (6.13) and(6.14) become ineffective for proving uniform bounds, since for any fixed τ < ∞ the value Γ( τ ) becomes too small in the limit ε → . This is due to the distancethat some row functions S ( ε ) x , with | x − δ | ≤ ε , have from all the other rowfunctions.Let m = m ( ε ) denote the solution of the QVE corresponding to S ( ε ) . We willnow show that, even though m (0) is uniformly bounded, the B -norm of m ( ε ) diverges as ε approaches zero for certain parameters λ and δ .The solution m = m ( ε ) has the form m x ( z ) = − z + ϕ ( z ) r x + ψ ( z ) s x . Here, the two functions ϕ ( ε ) = ϕ = (cid:104) s, m (cid:105) , ψ ( ε ) = ψ = (cid:104) r, m (cid:105) : H → H satisfythe coupled equations ϕ ( z ) = − (cid:90) [0 , s x d xz + ϕ ( z ) r x + ψ ( z ) s x ψ ( z ) = − (cid:90) [0 , r x d xz + ϕ ( z ) r x + ψ ( z ) s x . (11.7)In the parameter regime λ ≥ and δ ≤ / the support of the generatingdensity of m (0) consists of three disjoint intervals, supp v (0) = supp ϕ (0) = supp ψ (0) = [ − β , − α ] ∪ [ − α , α ] ∪ [ α , β ] . ε decreases the average generatingdensity remains bounded. The absolute value of thesolution as a function of x at a fixed value τ insidethe gap of the limiting generating density has a blowup.Inside the gap ( α , α ) the norm (cid:107) m ( ε ) (cid:107) di-verges as ε ↓ .This can be seen in-directly, by utilizingTheorem 2.6. Wewill now sketch an ar-gument, which showsthat assuming a uni-form bound on m leadsto a contradiction. Sup-pose there were an ε -independent bound onthe uniform norm. Then a local version of Theorem 2.6 would be applicableand the generating density v ( ε ) of m ( ε ) could approach zero only in the specificways described in that theorem. Instead, the average generating density (cid:104) v ( ε ) (cid:105) takes small non-zero values along the whole interval ( α , α ) , as we explainbelow. This contradicts the assertion of the theorem.In fact, a stability analysis of the two equations (11.7) for ϕ ( ε ) and ψ ( ε ) shows that they are uniformly Lipshitz-continuous in ε . In particular, for τ wellinside the interval ( α , α ) we have Im ϕ ( ε ) ( τ ) + Im ψ ( ε ) ( τ ) ≤ C ε .
Thus, the average generating density takes small values here as well, (cid:104) v ( ε ) ( τ ) (cid:105) ≤ C ε . On the other hand, Im ϕ and Im ψ do not vanish on ( α , α ) . Their supportscoincide with the support of the generating density, v ( ε ) . By Theorem 2.11 thissupport is a single interval for all ε > and by the continuity of ϕ and ψ in ε ,every point τ ∈ ( − β , − α ) ∪ ( α , β ) is contained in this interval in the limit ε ↓ .This example demonstrates that certain features of the solution of the QVEcannot be expected to be stable under smoothing of the corresponding operator S . Among these features are gaps in the support of the generating density, aswell as the universal shapes described by Theorem 2.6. z = 0 when a = 0 and assumptionB1 In the case a = 0 , the point z = 0 plays a special role in the QVE. It is the onlyplace where m ( z ) may become unbounded even in the L -sense (cf. (6.31)). Inthis section we give two simple examples which exhibit different types of blow-ups at z = 0 . Moreover, we motivate the assumption B1 by showing that itcorresponds to a necessary condition for the solution to remain bounded in astable way at z = 0 when the dimension of X is finite.Suppose a = 0 . The assumption B1 is designed to prevent divergencies inthe solution at the origin of the complex plane. These divergencies are caused127y the structure of small values of the kernel S xy . In Section 6.2 we saw thatat z = 0 the QVE reduces to v x (cid:90) X S xy v y π (d y ) = 1 , x ∈ X , (11.8)where v x = Im m x (0) . Thus the boundedness of m ( z ) for small | z | is related tothe solvability of (11.8). There is an extensive literature on (11.8) that datesback at least to [Sin64].In the discrete setup, with X := { , . . . , N } and π ( { i } ) := N − , the solv-ability of (11.8) is equivalent to the scalability (cf. Definition A.2) of the ma-trix S = ( s ij ) Ni,j =1 , with non-negative entries s ij := N − S ij . We refer to Ap-pendix A.3 for a discussion of various issues related to scalability. Theorem A.4below shows that the discrete QVE has a unique bounded solution if and only ifthe matrix S is fully indecomposable. This bound may deteriorate in N . How-ever, if S is block fully indecomposable (the property B1 ), then the bound onthe solution depends only on the number of blocks (cf. (6.38) and Lemma 6.10).Let us go back to the continuum setting. If assumption B1 is violated, thegenerating measure may have a singularity at z = 0 . In fact, there are twotypes of divergencies that may occur. Either the generating density exists ina neighborhood of τ = 0 and has a singularity at the origin, or the generatingmeasure has a delta-component at the origin. Both cases can be illustratedusing the × -block operator with the integral kernel (11.2).The latter case occurs if the kernel S xy contains a rectangular zero-blockwhose circumference is larger than . For S from (11.2) this means that δ > / .Expanding the corresponding QVE for small values of z reveals v x (d τ ) = π δ − δ { x ≤ δ } δ (d τ ) + O (1)d τ . The components of the generating measure with x ∈ [0 , δ ] assign a non-zerovalue to the origin.The case of a singular, but existing generating density can be seen from thesame example, (11.2), with the choice δ = 1 / . From an expansion of the QVEat small values of z we find for the generating density: v x ( τ ) = (2 λ ) − / √ | τ | − / (cid:8) x ≤ (cid:9) + O (1) . The blow-up at z = 0 has a simple interpretation in the context of randommatrix theory. It corresponds to an accumulation of eigenvalues at zero. If thegenerating density assigns a non-zero value to the origin, a random matrix withthe corresponding S as its variance matrix (cf. Definition 3.1) will have a kernel,whose dimension is a finite fraction of the size N of the matrix.Assumption B1 excludes the above examples. In general, it ensures that adiscretized version, of dimension K , of the original continuous problem (11.8)has a unique bounded and stable solution by the part (i) of Theorem A.4. Thebounded discrete solution is then used in Section 6.2 to argue that also thecontinuous problem has a bounded solution by using a variational formulation(11.8). 128 a For most of our analysis the function a ∈ B has played a secondary role. How-ever, even for the simplest operator S the addition of a non-constant a to theQVE without a can alter the solution significantly. Indeed, let us consider thesimplest case S xy = 1 , so that A1-3 hold trivially. Since (cid:104) w, Sw (cid:105) = (cid:104) w (cid:105) , forany w ∈ L , S satisfies also B2 , and thus Lemma 6.7 yields a uniform L -bound sup z ∈ H (cid:107) m ( z ) (cid:107) (cid:46) . Since ( Sm ( z )) x = (cid:104) m ( z ) (cid:105) for any x , we obtain a closedscalar integral equation for the average of m ( z ) (cid:104) m ( z ) (cid:105) = (cid:90) X π (d x ) z + a x + (cid:104) m ( z ) (cid:105) , (11.9)by integrating the QVE. If a is piecewise / -Hölder regular in the sense of(2.25), then Theorem 6.4 yields a uniform bound ||| m ||| R (cid:46) (see Remark 6.5).In particular, Theorem 2.6 applies.In the random matrix context (11.9) determines the asymptotic density ofstates of a deformed Wigner matrix , H = A + W , (11.10)where W is an N -dimensional Wigner matrix, and A is a self-adjoint non-random matrix satisfying Spec( A ) = { a i : 1 ≤ i ≤ N } , in the limit N → ∞ (cf.[Pas72]).In the special case, that N is an even integer and A has only two eigenvalues ± α , both of degeneracy N/ , i.e., a k := (cid:40) − α when ≤ k ≤ N/ α when N/ ≤ k ≤ N , (11.11)the equation (11.9) can be reduced to a single cubic polynomial for (cid:104) m ( z ) (cid:105) . In[BH98] this matrix model (11.10) was analyzed and the authors demonstratedthat the asymptotic density of states may exhibit a cubic root cusp for somevalues of the parameter α . The cubic root singularity seems natural in thespecial case (11.11) as (cid:104) m ( z ) (cid:105) satisfies a cubic polynomial. If the range of a contains p ∈ N distinct values, then (11.9) can be reduced to a polynomial ofdegree p + 1 . Our results, however, show that in spite of this arbitrary highdegree, the worst possible singularity is cubic, and the possible shapes of thedensity of states are described by Theorem 2.6, as long as a is sufficiently regular. By choosing X := { , . . . , N } and π ( { i } ) := N − for some N ∈ N the QVE (2.4)takes the form − m i = z + a i + 1 N N (cid:88) j =1 S ij m j , i = 1 , . . . , N , (11.12) 129nd hence this discrete vector equation is covered by our analysis. Alternatively,we may treat (11.12) in the continuous setup (11.1) by defining a function a : [0 , → R and the integral kernel of S on [0 , by a ( x ) := N (cid:88) i =1 a i χ i ( x ) , and S ( x, y ) := N (cid:88) i,j =1 S ij χ i ( x ) χ j ( y ) , (11.13)respectively, with the auxiliary functions χ i : [0 , → { , } , i = 1 , . . . , N , givenby χ i ( x ) := (cid:8) N x ∈ [ i − , i ) (cid:9) . In order to distinguish between discrete and continuous quantities we haveadapted in this section a special convention by writing the continuous variable x in the parenthesis and not as a subscript. Since the continuous QVE conservesthe block structure, and both the discrete and continuous QVEs have uniquesolutions m = ( m i ) Ni =1 and m = ( x (cid:55)→ m ( x )) , respectively, by Theorem 2.1, weconclude that these solutions are related by m ( z ; x ) = N (cid:88) i =1 m i ( z ) χ i ( x ) . (11.14)This re-interpretation of a discrete QVE as a continuous one is convenientwhen comparing different discrete QVEs of non-matching dimensions N . Forexample, the convergence of a sequence of QVEs generated by a smooth function α : [0 , → R and a symmetric smooth function σ : [0 , → [0 , ∞ ) , through a i := α (cid:16) iN (cid:17) , and S ij := σ (cid:16) iN , jN (cid:17) , i, j = 1 , . . . , N , can be handled this way. Indeed, if m solves the discrete QVE then the functions m defined through the right hand side of (11.14) converge to the solution of thecontinuous QVE with a ( x ) = α ( x ) and S ( x, y ) := σ ( x, y ) as N → ∞ .In particular, if the continuum operator satisfies A3 and B2 , or merely B1 in the case α = 0 (all other assumptions are automatic in this case), thenthe convergence of the generating densities is uniform and the support of thegenerating density is a single interval for large enough N . This is a consequenceof the stability result, Theorem 2.12, more precisely of Remark 2.13 following itand of the fact that the limiting operator S is block fully indecomposable, andthe knowledge about the shape of the generating density from Theorem 2.6 andTheorem 2.11.We also have the following straightforward dimensional reduction. Supposethere exists a partition I of the first N integers, and numbers ( (cid:98) S IJ ) I,J ∈I and ( (cid:98) a I ) I ∈I , indexed by the parts, such that for every I, J ∈ I and i ∈ I , (cid:88) j ∈ J S ij = | J | (cid:98) S IJ , and a i = (cid:98) a I . m ( z ) is piecewise constant on the parts of I , i.e., there exist numbers (cid:98) m ( z ) = ( (cid:98) m I ( z )) I ∈I , such that m i ( z ) = (cid:98) m I ( z ) , for every i ∈ I . The numbers (cid:98) m ( z ) solve the |I| -dimensional reduced QVE, − (cid:98) m I ( z ) = z + (cid:98) a I + (cid:88) J ∈I | J | N (cid:98) S IJ (cid:98) m J ( z ) . Here the right hand side can be written in the standard form (2.4) by identifying X = I and (cid:98) π ( J ) = | J | /N . In the special case where the matrix S = ( S ij ) Ni,j =1 has constant row sums, N − (cid:80) j S ij = 1 , and a = 0 , the reduced QVE is one-dimensional, and is solved by the Stieltjes transform of the Wigner semicirclelaw (1.4)The dimension reduction argument generalizes trivially to more abstractsetups. Indeed, we have already used such a reduction in Section 11.2, wherewe reduced the analysis of the infinite dimensional QVE, with an integral kernel S xy defined in (11.2), to the study of the two-dimensional QVE (11.4). We will now discuss how all possible shapes of the generating density fromTheorem 2.6 can be seen in the simple example of the × -block operator S ,defined in (11.2), by choosing the parameters λ and δ appropriately. For thechoice of parameters λ > and δ = δ c ( λ ) with δ c ( λ ) := ( λ − λ − λ + 15 λ − , the generating density exists everywhere and its support is a single interval.In the interior of this interval the generating density has exactly two zeros atsome values τ c and − τ c . The shape of the generating density at these zerosin the interior of its own support is a cubic cusp, represented by the shapefunction lim ρ ↓ ρ Ψ min ( ω/ρ ) = 2 / | ω | / (cf. Definition 2.5). If we increase δ above δ c ( λ ) , then the zeros of the generating density disappear. The support isa single interval with local minima close to τ c and − τ c and the shape aroundthese minima is described by ρ Ψ min ( · /ρ ) for some small positive ρ . Finally, ifwe decrease δ slightly below δ c ( λ ) a gap opens in the support. Now the supportof the generating density consists of three disjoint intervals and the shape of thegenerating density at the two neighboring edges is described by ∆ / Ψ edge ( · / ∆) ,where ∆ (cid:28) is the size of the gap. The different choices of δ are illustrated inFigure 11.3. 131igure 11.3: Decreasing δ from its critical value δ c opens a gap in the sup-port of the average generating density. Increasing delta lifts the cubic cuspsingularity. 132 ppendix A Appendix
The following simple comparison relations are used in the proof of Proposi-tion 10.1 when Im z (cid:54) = 0 and Re z is close to a local minimum of the generatingdensity. Corollary A.1 (Scaling relations) . Suppose the assumptions of Theorem 2.6are satisfied. There exists a positive threshold ε ∼ such that for the set of localminima M , defined in (2.17) , and any η ∈ (0 , ε ] , the average generating densityhas the following growth behavior close to the points in M : (a) Support around an edge:
At the edges α i , β i − with i = 2 , . . . , K (cid:48) , (cid:10) Im m ( α i + ω + i η ) (cid:11) ∼ (cid:10) Im m ( β i − − ω + i η ) (cid:11) ∼ ( ω + η ) / ( α i − β i − + ω + η ) / , ω ∈ [0 , ε ] . (b) Inside a gap:
Between two neighboring edges β i − and α i with i =2 , . . . , K (cid:48) , (cid:10) Im m ( τ + i η ) (cid:11) ∼ η ( α i − β i − + η ) / × (cid:18) τ − β i − + η ) / + 1( α i − τ + η ) / (cid:19) , τ ∈ [ β i − , α i ] . (c) Support around an extreme edge:
Around the extreme points α and β K (cid:48) of supp v : (cid:10) Im m ( α + ω + i η ) (cid:11) ∼ (cid:10) Im m ( β K (cid:48) − ω + i η ) (cid:11) ∼ ( ω + η ) / , ω ∈ [0 , ε ] ; η ( | ω | + η ) / , ω ∈ [ − ε, . Close to a local minimum:
In a neighborhood of the local minima { γ k } in the interior of the support of the generating density, (cid:10) Im m ( γ k + ω + i η ) (cid:11) ∼ (cid:104) v ( γ k ) (cid:105) + ( | ω | + η ) / , ω ∈ [ − ε, ε ] . All constants hidden behind the comparison relations depend on the parameters ρ , L , (cid:107) S (cid:107) L → B and Φ . Proof.
The results follow by combining Theorem 2.6 and the Stieltjes trans-form representation of the solution of QVE. We start with the claim about thegrowth behavior around the points { γ k } . By the description of the shape of thegenerating density in Theorem 2.6 and because of Ψ min ( λ ) ∼ min { λ , | λ | / } (cf. (2.14b)), we have for small enough ε ∼ : (cid:104) v ( γ k + ω ) (cid:105) ∼ ρ k + min (cid:8) ω /ρ k , | τ | / (cid:9) ∼ ρ k + | ω | / , ω ∈ [ − ε, ε ] . The constant ρ k is comparable to (cid:104) v ( γ k ) (cid:105) by (2.20c). Thus, we find (cid:10) Im m ( γ k + ω +i η ) (cid:11) = 1 π (cid:90) ∞−∞ η (cid:104) v ( τ ) (cid:105) d τη + ( γ k + ω − τ ) ∼ (cid:104) v ( γ k ) (cid:105) + (cid:90) ε − ε η | τ | / d τη + ( ω − τ ) , for ω ∈ [ − ε, ε ] . The claim follows because the last integral is comparable to ( | ω | + η ) / for any ε ∼ .Let us now consider the case, in which an edge is close by. We treat only thecase of a right edge, i.e., the vicinity of β i for i = 1 , . . . , K (cid:48) . For the left edgethe argument is the same. Here, Theorem 2.6 and Ψ edge ( λ ) ∼ min { λ / , λ / } (cf. (2.14a)) imply for small enough ε ∼ : (cid:104) v ( β i − ω ) (cid:105) ∼ min { ∆ − / ω / , ω / } , ω ∈ [0 , ε ] . The positive constant ∆ is comparable to the gap size, ∆ ∼ α i +1 − β i , if β i isnot the rightmost edge, i.e., i (cid:54) = K (cid:48) . In case i = K (cid:48) , we have ∆ ∼ . Let us set (cid:101) ε := ε in case i = K (cid:48) , and (cid:101) ε := min { ε, ( α i +1 − β i ) / } otherwise. Then we find (cid:10) Im m ( β i + ω + i η ) (cid:11) = 1 π (cid:90) ∞−∞ η (cid:104) v ( τ ) (cid:105) d τη + ( β i + ω − τ ) ∼ η (cid:90) ε min { ∆ − / τ / , τ / } η + ( ω + τ ) d τ , ω ∈ (cid:2) − ε, (cid:101) ε (cid:3) . The contribution to the integral in the middle, coming from the other side α i +1 of the gap ( β i , α i +1 ) , is not larger than the last expression, because the growthof the average generating density is the same on both sides of the gap. For thelast integral we find η (cid:90) ε min { ∆ − / τ / , τ / } η + ( ω + τ ) d τ ∼ η (∆ + η ) / ( ω + η ) / , ω ∈ (cid:2) , (cid:101) ε (cid:3) ;( | ω | + η ) / (∆ + | ω | + η ) / ω ∈ (cid:2) − ε, (cid:3) . This holds for any ε ∼ and thus the claim of the lemma follows.134 .1 Proofs of auxiliary results in Chapter 4 Proof of Lemma 4.6.
Recall that T is a generic bounded symmetric operatoron L = L ( X ; C ) that preserves non-negative functions. Moreover, the followingis assumed: ∃ h ∈ L s.t. (cid:107) h (cid:107) = 1 , T h ≤ h , and ε := inf x ∈ X h x > . (A.2)We show that (cid:107) T (cid:107) L → L ≤ . Let us derive a contradiction by assuming (cid:107) T (cid:107) L → L > . We have T n h ≤ h , ∀ n ∈ N . (A.3)Indeed, T h ≤ h is true by definition, and (A.3) follows by induction.Now, the property (cid:107) T (cid:107) L → L > would imply ∃ u ∈ B s.t. (cid:107) u (cid:107) = 1 , u ≥ , and (cid:104) u, T u (cid:105) > . Since T is positive, (cid:104) u, T u (cid:105) ≤ (cid:104)| u | , T | u |(cid:105) , so we may assume u ≥ . Moreover,by standard density arguments we may assume (cid:107) u (cid:107) < ∞ as well.Since (cid:104) u, T u (cid:105) > , we obtain, by inserting u -projections between the T ’s: (cid:104) u, T n u (cid:105) ≥ (cid:104) u, T u (cid:105)(cid:104) u, T n − u (cid:105) ≥ · · · ≥ (cid:104) u, T u (cid:105) n → ∞ as n → ∞ . (A.4)The contradiction follows now by combining (A.3) and (A.4): (cid:104) h, u (cid:105) ≥ (cid:104) T n h, u (cid:105) = (cid:104) h, T n u (cid:105) ≥ (cid:104) h, u (cid:105)(cid:104) u, T n u (cid:105) . (A.5)The left hand side is less than (cid:107) h (cid:107) (cid:107) u (cid:107) = 1 . On the other hand, since h ≥ ε , u ≥ and (cid:107) u (cid:107) = 1 we have (cid:104) h, u (cid:105) > . Thus (A.4) implies that the right sideof (A.5) approaches infinity as n grows. A.2 Proofs of auxiliary results in Chapter 5
Proof of Lemma 5.7.
First we note that h is bounded away from zero by h = T h ≥ ε (cid:90) X π (d x ) h x . (A.6)Let u be orthogonal to h in L . Then we compute (cid:10) u, (1 ± T ) u (cid:11) = 12 (cid:90) π (d x ) (cid:90) π (d y ) T xy (cid:32) u x (cid:114) h y h x ± u y (cid:115) h x h y (cid:33) ≥ ε (cid:90) π (d x ) (cid:90) π (d y ) h x h y (cid:32) u x h y h x + u y h x h y ± u x u y (cid:33) = ε Φ (cid:90) π (d x ) u x , T xy ≥ ε ≥ ε h x h y / Φ for almost all x, y ∈ X .Now we read off the following two estimates: (cid:90) X π (d x ) u x ( T u ) x ≤ (cid:16) − ε Φ (cid:17) (cid:107) u (cid:107) , (cid:90) X π (d x ) u x ( T u ) x ≥ − (cid:16) − ε Φ (cid:17) (cid:107) u (cid:107) . This shows the gap in the spectrum of the operator T . Proof of Lemma 5.10.
In order to prove the claim (5.42) we will show (cid:107) ( U − T ) w (cid:107) ≥ c θ Gap( T ) (cid:107) w (cid:107) , θ := | − (cid:107) T (cid:107) (cid:104) h, U h (cid:105)| , (A.7)for all w ∈ L and for some numerical constant c > . To this end, let us fix w with (cid:107) w (cid:107) = 1 . We decompose w according to the spectral projections of T , w = (cid:104) h, w (cid:105) h + P w , (A.8)where P is the projection onto the orthogonal complement of t . During thisproof we will omit the lower index of all norms, since every calculation is in L . We will show the claim in three separate regimes:(i) (cid:107) P w (cid:107) ≥ θ ,(ii) (cid:107) P w (cid:107) < θ and θ ≥ (cid:107) P U h (cid:107) ,(iii) (cid:107) P w (cid:107) < θ and θ < (cid:107) P U h (cid:107) .In the regime (i) the triangle inequality yields (cid:107) ( U − T ) w (cid:107) ≥ (cid:107) w (cid:107) − (cid:107) T w (cid:107) = 1 − (cid:112) |(cid:104) h, w (cid:105)| (cid:107) T (cid:107) + (cid:107) T P w (cid:107) . We use the simple inequality, − √ − τ ≥ τ / , valid for every τ ∈ [0 , , andfind (cid:107) ( U − T ) w (cid:107) ≥ − |(cid:104) h, w (cid:105)| (cid:107) T (cid:107) − (cid:107) T P w (cid:107) ≥ − |(cid:104) h, w (cid:105)| (cid:107) T (cid:107) − ( (cid:107) T (cid:107) − Gap( T )) (cid:107) P w (cid:107) = 1 − (cid:107) T (cid:107) + (cid:0) (cid:107) T (cid:107) − Gap( T ) (cid:1) Gap( T ) (cid:107) P w (cid:107) . (A.9)The definition of the first regime implies the desired bound (A.7).In the regime (ii) we project the left hand side of (A.7) onto the h -direction,(A.10) (cid:107) ( U − T ) w (cid:107) = (cid:107) (1 − U ∗ T ) w (cid:107) ≥ |(cid:104) h, (1 − U ∗ T ) w (cid:105)| . Using the decomposition (A.8) of w and the orthogonality of h and P w , weestimate further: |(cid:104) h, (1 − U ∗ T ) w (cid:105)| ≥ |(cid:104) h, w (cid:105)| | − (cid:107) T (cid:107)(cid:104) h, U ∗ t (cid:105)| − |(cid:104) h, U ∗ T P w (cid:105)|≥ |(cid:104) h, w (cid:105)| θ − (cid:107) P U h (cid:107)(cid:107)
P w (cid:107) . (A.11) 136ince θ ≤ and by the definition of the regime (ii) we have |(cid:104) h, w (cid:105)| = 1 −(cid:107) P w (cid:107) ≥ − θ/ ≥ / and (cid:107) P U h (cid:107)(cid:107)
P w (cid:107) ≤ θ/ . Thus, we can combine(A.10) and (A.11) to (cid:107) ( U − T ) w (cid:107) ≥ θ . Finally, we treat the regime (iii). Here, we project the left hand side of (A.7)onto the orthogonal complement of h and get(A.12) (cid:107) ( U − T ) w (cid:107) ≥ (cid:107) P ( U − T ) w (cid:107) ≥ |(cid:104) h, w (cid:105)|(cid:107) P U h (cid:107) − (cid:107) P ( U − T ) P w (cid:107) , where we inserted the decomposition (A.8) again. In this regime we still have |(cid:104) h, w (cid:105)| ≥ / , and we continue with(A.13) |(cid:104) h, w (cid:105)|(cid:107) P U h (cid:107) − (cid:107) P ( U − T ) P w (cid:107) ≥ (cid:107) P U h (cid:107) − (cid:107) P w (cid:107) ≥ θ / . In the last inequality we used the definition of the regime (iii). Combining(A.12) with (A.13) yields (cid:107) ( U − T ) w (cid:107) ≥ θ , after using (cid:107) h (cid:107) = 1 in (A.7) to estimate θ ≤ . A.3 Scalability of matrices with non-negative en-tries
In this appendix we provide some background material for Sections 6.2 and11.3. We start by introducing some standard terminology related to matriceswith non-negative entries. First, let us denote [ k, l ] := { k, k + 1 , . . . , l } , for anyintegers k ≤ l . We use the shorthand [ n ] := [1 , n ] , and denote the | I | × | J | -submatrix A ( I, J ) := ( a ij ) i ∈ I,j ∈ J , for any non-empty sets I, J ⊂ [ n ] . The set of all permutations of [ n ] is denotedby S ( n ) , and we say that P = ( p ij ) ni,j =1 is a permutation matrix, if its entriesare determined by some permutation σ ∈ S ( n ) through p ij = δ σ ( i ) ,j . Definition A.2.
Let A = ( a ij ) ni,j =1 be a square matrix with non-negativeentries, a ij ≥ . Then:(i) A is scalable if there exist two diagonal matrices D and D (cid:48) with positiveentries, such that the scaled matrix DAD (cid:48) is doubly stochastic.(ii) A is uniquely scalable if it is scalable and the pair of diagonal matrices ( D , D (cid:48) ) is unique up to a scalar multiple.137iii) A has total support if there exists a set of permutations T ⊂ S ( n ) , suchthat a ij = 0 if and only if (cid:88) σ ∈ T δ σ ( i ) ,j = 0 , ∀ i, j = 1 , . . . , n . (A.14)(iv) A is decomposable if it is not fully indecomposable, i.e., there exist twonon-empty subsets I, J ⊂ [ n ] such that A ( I, J ) = and | I | + | J | ≥ n . (A.15)We remark that all these four properties of A are invariant under the trans-formations A (cid:55)→ PAQ , where P and Q are arbitrary permutation matrices.The defining condition (A.14) for matrices A with total support means that A shares its zero entries with some doubly stochastic matrix. This fact followsfrom Birkhoff-von Neumann theorem which asserts that the doubly stochasticmatrices are exactly the convex combinations of permutation matrices.Besides the elementary properties stated in Proposition 6.9 the fully in-decomposable (FID) matrices are also building blocks for matrices with totalsupport. Indeed, Theorem 4.2.8 of [BR91] asserts: Theorem A.3. If A has total support then there exist two permutation matrices P and Q such that PAQ is a direct sum of FID matrices.
Consider the QVE with a = 0 at z = 0 in the discrete setup ( X , π ) =([ n ] , n − | · | ) . From (6.26) we read off that this QVE has a unique solutionof the form m (0) = i v provided the matrix S , with entries s ij := n − S ij ,is scalable such that VSV is doubly stochastic for the diagonal matrix V =diag( v , . . . , v n ) . This observation together with the equivalence of (i) and (iii)in the following theorem shows that in the discrete setup the assumption B1 from Chapter 6, with the trivial blocks K = n , is actually optimal in the part(i) of Theorem 6.1. Theorem A.4 (Scalability and full indecomposability) . For a symmetric irre-ducible matrix A with non-negative entries the following are equivalent: (i) A is uniquely scalable, with D (cid:48) = D in Definition A.2; (ii) Every sufficiently small perturbation of A is scalable, i.e., there exists aconstant ε > such that any symmetric matrix A (cid:48) , with non-negativeentries, satisfying max i,j | a ij − a (cid:48) ij | ≤ ε , is scalable; (iii) A is fully indecomposable. The proof of Theorem A.4 relies on the following fundamental result.
Theorem A.5 ([SK67]) . A square matrix A with non-negative entries is scalable if and only if it has a total support; (ii) uniquely scalable if and only if it is fully indecomposable.Moreover, if A is scalable, then the doubly stochastic matrix DAD (cid:48) , fromDefinition A.2, is unique.
For the proof of Theorem A.4 we need also the following representation.
Lemma A.6 (Scalable symmetric matrices) . Suppose A = ( a ij ) ni,j =1 is an irre-ducible symmetric matrix with non-negative entries. If A has a total support butis not fully indecomposable, then n is even, and there exists an n/ -dimensionalsquare matrix B , and a permutation matrix P , such that A = P (cid:20) T (cid:21) P − . (A.16) Proof of Lemma A.6.
Since A is not FID there exists by Definition A.2 twonon-empty subsets I, J ⊂ [ n ] , such that (A.15) holds. Let us relabel the indicesso that I = [1 , n ] , J = [ n , n ] , for some ≤ n ≤ n ≤ n ≤ n . The relabellingcorresponds to the conjugation by the permutation matrix P in (A.16). Bydefinition (A.15) of I and J we have P − AP = A A A T14 A T24 A T34 A , (A.17)where the blocks correspond to the four intervals I = [1 , n ] , I = [ n + 1 , n ] , I = [ n + 1 , n ] , and I = [ n + 1 , n ] , respectively. In the case, n k +1 = n k theinterval I k is interpreted to be empty.Now we show that | I | ≤ | I | . Indeed, P − AP has a zero block of size | I | × ( n − | I | ) . No permutation matrix can have such a zero block if | I | > | I | .As A , and thus also P − AP , has total support, the defining property (A.14)could not hold for A if | I | > | I | were true.By definitions, | I | = | I | + | I | and | J | = | I | + | I | , and by assumption | I | + | J | ≥ n . Since n = | I | + | I | + | I | + | I | , we conclude | I | ≥ | I | .Since | I | = | I | the submatrix A is square. This implies that σ ( I ) = I for the permutations σ ∈ T in the representation (A.14). This is equivalent to σ ( I ∪ I ∪ I ) = I ∪ I ∪ I , and thus A = , A = , and A = .But now we see that I and I must be empty intervals, otherwise A wouldbe an independent block of A , and thus A would not be irreducible. Since I = I = ∅ , we conclude I = J . But this leaves us with the representation(A.16) with B := A . Proof of Theorem A.4.
The equivalence of (i) and (iii) almost follows fromthe part (ii) of Theorem A.5. We are only left to exclude the possibility that A is not FID since it is not uniquely scalable for general pairs ( D , D (cid:48) ) , but is139ctually uniquely scalable in the more restricted class of ’diagonal solutions’ forwhich D (cid:48) = D holds.To this end we show that if a symmetric and irreducible matrix A withnon-negative entries is scalable, then we may always choose D (cid:48) = D . First werecall that the doubly stochastic matrix B := DAD (cid:48) is unique according toTheorem A.5. Since A is symmetric, D (cid:48) AD = B T is also doubly stochastic. ByTheorem A.5 we hence have D (cid:48) AD = DAD (cid:48) . We may write this in terms ofthe ratios ρ i = d (cid:48) ii /d ii , as ρ i = ρ j , whenever a ij > . (A.18)Pick any i (cid:54) = j . Since, A is irreducible, there exists a sequence ( k s ) (cid:96)s =0 , (cid:96) ≤ n ,of indices such that k = i , k (cid:96) = j , and a k s − k s > for every s = 1 , . . . , (cid:96) , thus ρ i = ρ j by (A.18). We conclude D (cid:48) = ρ D , and thus we may choose D (cid:48) = D byfurther scaling by a scalar.In order to prove the implication (iii) = ⇒ (ii), choose ε to be equal to thesmallest non-zero entry of A . It follows that the ε -perturbation A (cid:48) in (ii) has asmaller set of entries equal to zero than A . Thus with this choice of ε the zeroset of the perturbation A (cid:48) may only decrease. By Definition 2.9 A (cid:48) is thus alsoFID, and by Theorem A.5 A (cid:48) is scalable.In order to prove the last implication (ii) = ⇒ (iii), we assume that A isnot FID, and derive a contradiction by showing that the perturbed matrix, A (cid:48) := A + ε ∆ ( ij ) , ( ∆ ( ij ) ) kl := { { k, l } = { i, j } } , (A.19)does not have total support for all choices of ( i, j ) , regardless of how small ε > is chosen. We start by using Lemma A.6 to write A in the form A = (cid:20) T (cid:21) . (A.20)Here we have also relabelled the indices such that P = I in (A.16). Suppose thatwe turn one of the zero entries in the first n/ × n/ diagonal block non-zero,i.e., consider a perturbation (A.19), for some i, j ≤ n/ . We will show that theredoes not exist a subset T (cid:48) of permutations S ( n ) such that the representation(A.14), with T replaced by T (cid:48) , holds for A (cid:48) . Indeed, suppose that there is sucha set of permutations T (cid:48) . Since a ij > there must exist σ ∈ T (cid:48) such that σ ( i ) = j . This implies that [ n/ \ σ ([ n/ { k } , for some k ≤ n/ . Since σ is a surjection on { , . . . , n } there must exist l ≥ n/ such that σ ( l ) = k . In other words, there exists an entry ( l, k ) in thesecond diagonal block, l, k ≥ n/ , such that a (cid:48) lk = a lk > . Since this contradicts(A.20) and (A.19), we conclude that A (cid:48) does not have total support.140 .4 Variational bounds when Re z = 0 Proof of Lemma 6.8.
Applying Jensen’s inequality on the definition (6.28)of J η yields, J η ( w ) ≥ (cid:104) w, Sw (cid:105) − (cid:104) w (cid:105) + 2 η (cid:104) w (cid:105) . The lower bound shows that the functional J η is indeed well defined and takesvalues in ( −∞ , + ∞ ] . Evaluating J η on a constant function shows that it is notidentically + ∞ .Next we show that J η has a unique minimizer on the space L (cf. definition(6.27)) of positive integrable functions. As the first step, we show that we canrestrict our attention to functions, which satisfy the upper bound w ≤ /η . Tothis end, pick w ∈ L , such that the set { x : w x ≥ η − } has positive π -measure,and define the one parameter family of L -functions w ( τ ) := w − τ ( w − η − ) + , ≤ τ ≤ , where φ + := max { , φ } , φ ∈ R . It follows that w ( τ ) ≤ w (0) = w and J η ( w ( τ )) < ∞ for every τ ∈ [0 , . We will show that J η (cid:0) min( w, η − ) (cid:1) = J η ( w (1)) < J η ( w ) . (A.21)For this we compute dd τ J η ( w ( τ )) = − (cid:28)(cid:16) Sw ( τ ) + η − w ( τ ) (cid:17)(cid:0) w − η − (cid:1) + (cid:29) . (A.22)Since w ≥ and therefore Sw ≥ , the integrand is positive on the set of x where w x > /η . Thus, the derivative (A.22) is strictly positive for τ ∈ [0 , .We conclude that the minimizer must be bounded from above by η − .Now we use a similar argument to see that we may further restrict the searchof the minimizer to functions which satisfy also the lower bound w ≥ η/ (1 + η ) .To this end, fix w ∈ L satisfying J η ( w ) < ∞ and (cid:107) w (cid:107) ∞ ≤ η − . Suppose w < η/ (1 + η ) , on some set of positive π -measure, and set w ( τ ) := w + (cid:16) η η − w (cid:17) + τ , so that w = w (0) ≤ w ( τ ) , and J η ( w ( τ )) < ∞ , for every τ ∈ [0 , . Differentia-tion yields, dd τ J η ( w ( τ )) ≤ (cid:28)(cid:16) η + η − w ( τ ) (cid:17)(cid:16) η η − w (cid:17) + (cid:29) , where the term η − originates from (cid:107) Sw ( τ ) (cid:107) ≤ (cid:107) S (cid:107)(cid:107) w ( τ ) (cid:107) ≤ η − . Since η − + η = ( η/ (1 + η )) − , and w < η/ (1 + η ) on a positive set of positive measure,we again conclude that J η ( w (1)) < J η ( w ) .Consider now a sequence ( w ( n ) ) n ∈ N in L that satisfies lim n →∞ J η ( w ( n ) ) = inf w J η ( w ) and η η ≤ w ( n ) ≤ η . w ( n ) also constitutes a bounded sequence of L . Consequently, thereis a subsequence, denoted again by ( w ( n ) ) n ∈ N , that converges weakly to anelement w (cid:63) of L . This weak limit also satisfies η η ≤ w (cid:63)x ≤ η , ∀ x ∈ X . (A.23)In order to conclude that w (cid:63) is indeed a minimizer of J η we will show that J η is weakly continuous in L at all points w (cid:63) satisfying the bounds (A.23).To this end, we consider the three term constituting J η separately. Evidentlythe averaging u (cid:55)→ (cid:104) u (cid:105) is weakly continuous. For the quadratic form we firstcompute for any sequence w ( n ) converging weakly to w (cid:63) : (cid:12)(cid:12) (cid:104) w ( n ) , Sw ( n ) (cid:105) − (cid:104) w (cid:63) Sw (cid:63) (cid:105) (cid:12)(cid:12) ≤ (cid:0) (cid:107) w ( n ) (cid:107) + (cid:107) w (cid:63) (cid:107) (cid:1) (cid:107) S ( w ( n ) − w (cid:63) ) (cid:107) . (A.24)Since the L -norm is lower-semicontinuous and (cid:107) w (cid:63) (cid:107) ≤ (cid:107) w (cid:63) (cid:107) ≤ η − , we infer lim sup n →∞ (cid:12)(cid:12) (cid:104) w ( n ) , Sw ( n ) (cid:105) − (cid:104) w (cid:63) Sw (cid:63) (cid:105) (cid:12)(cid:12) ≤ η lim sup n →∞ (cid:13)(cid:13) S ( w ( n ) − w (cid:63) ) (cid:13)(cid:13) . Using the L -function, S x : X → [0 , ∞ ) , y (cid:55)→ S xy , we obtain: (cid:107) S ( w ( n ) − w (cid:63) ) (cid:107) = (cid:90) X π (d x ) (cid:12)(cid:12)(cid:12)(cid:90) X π (d y ) S xy ( w ( n ) − w (cid:63) ) y (cid:12)(cid:12)(cid:12) = (cid:90) X π (d x ) (cid:12)(cid:12) (cid:104) S x ( w ( n ) − w (cid:63) ) (cid:105) (cid:12)(cid:12) . Here the weak convergence of w ( n ) to w (cid:63) implies h ( n ) x := |(cid:104) S x ( w ( n ) − w (cid:63) ) (cid:105)| → for each x separately. The uniform bound | h ( n ) x | ≤ (cid:107) S x (cid:107) (cid:107) w ( n ) − w (cid:63) (cid:107) ≤ (cid:107) w ( n ) (cid:107) − (cid:107) w (cid:63) (cid:107) ) (cid:107) S (cid:107) → B , and the dominated convergence then yield: (cid:90) X π (d x ) (cid:12)(cid:12) (cid:104) S x ( w ( n ) − w (cid:63) ) (cid:105) (cid:12)(cid:12) = (cid:90) X π (d x ) h ( n ) x → , as n → ∞ . Hence the last term of (A.24) converges to zero as n goes to infinity, and wehave shown that the quadratic form is indeed weakly continuous at w (cid:63) .Finally, we show that also the logarithmic term is weakly continuous at w (cid:63) .Applying Jensen’s inequality yields (cid:12)(cid:12) (cid:104) log w ( n ) (cid:105) − (cid:104) log w (cid:63) (cid:105) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:68) log (cid:16) w ( n ) w (cid:63) (cid:17)(cid:69)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) log (cid:68) w ( n ) w (cid:63) (cid:69)(cid:12)(cid:12)(cid:12) , where the last average converges to by the assumed weak convergence of w ( n ) to w (cid:63) and since /w (cid:63) ∈ L by the lower bound in (A.23).We have proven the existence of a positive minimizer w (cid:63) ∈ L that satisfies(A.23). In order to see that w (cid:63)x = v x (i η ) for a.e. x ∈ X we evaluate a derivativeof J η ( w (cid:63) + τ h ) | τ =0 for an arbitrary h ∈ B . This derivative must vanish by thedefinition of w (cid:63) , and therefore ( Sw (cid:63) ) x + η − w (cid:63)x = 0 , for π -a.e. x ∈ X . (A.25) 142ince Sw , with w ∈ L , is insensitive to changing the values of w x , for x ∈ I ,whenever I ⊆ X is of measure zero, we may modify w (cid:63) on the zero measureset where the equation of (A.25) is not satisfied, so that the equality holdseverywhere. Since (A.25) equals QVE at z = i η Theorem 2.1 implies that(A.25) has v (i η ) as the unique solution. We conclude that w (cid:63)x = v x (i η ) for a.e. x ∈ X . Proof of Lemma 6.10.
Since Z is FID, the exists by the part (ii) of Propo-sition 6.9 a permutation σ of the first K integers, such that (cid:101) Z = ( (cid:101) Z ij ) Ki,j =1 , (cid:101) Z ij := Z iσ ( j ) , has a positive main diagonal, i.e., (cid:101) Z ii = 1 for every i . Let us define the convexfunction Λ : (0 , ∞ ) → R , by Λ( τ ) := ϕK τ + log 1 τ , where ϕ > and K ∈ N are from B2 . Clearly, lim τ →∞ Λ( τ ) = ∞ and lim τ → Λ( τ ) = ∞ . In particular, Λ( τ ) ≥ Λ − , (A.26)where | Λ − | (cid:46) , since ϕ and K are considered as model parameters,Using (cid:101) Z ii = 1 and w i (cid:101) Z ij w σ ( j ) ≥ in the definition (6.35) of (cid:101) J ( w ) , we obtain (cid:88) i Λ( w i w σ ( i ) ) ≤ (cid:88) i (cid:16) ϕK w i (cid:101) Z ii w σ ( i ) − log (cid:2) w i w σ ( i ) (cid:3)(cid:17) + ϕK (cid:88) i (cid:54) = j w i (cid:101) Z ij w σ ( j ) = (cid:101) J ( w ) . (A.27)Combining the assumption (cid:101) J ( w ) ≤ Ψ with the lower bounds (A.26) of Λ yields w k w σ ( k ) ∼ , ≤ k ≤ K . (A.28)Using (A.26) together with (A.27) and the hypothesis of the lemma, (cid:101) J ( w ) ≤ Ψ ,we obtain an estimate for the off-diagonal terms as well: ϕK (cid:88) i (cid:54) = j w i (cid:101) Z ij w σ ( j ) ≤ (cid:101) J ( w ) − (cid:88) i Λ( w i w σ ( i ) ) ≤ Ψ + K | Λ − | . (A.29)Since we consider ( ϕ, K, Ψ) as model parameters, the bounds (A.28) and (A.29)together yield M ij := w i (cid:101) Z ij w σ ( j ) (cid:46) . (A.30)This would imply the claim of the lemma, max i w i (cid:46) , provided we would have (cid:101) Z ij (cid:38) for all i, j . To overcome this limitation we compute the ( K − -th143ower of the matrix M formed by the components (A.30). This way we get touse the FID property of Z : ( M K − ) ij = (cid:88) i ,...,i K − w i (cid:101) Z ii w σ ( i ) w i (cid:101) Z i i w σ ( i ) w i × (cid:101) Z i i w σ ( i ) · · · w i K − (cid:101) Z i K − j w σ ( j ) ≥ (cid:16) min k w k w σ ( k ) (cid:17) K − ( (cid:101) Z K − ) ij w i w σ ( j ) . (A.31)Since Z is FID also (cid:101) Z is FID, and therefore min Ki,j =1 ( (cid:101) Z K − ) ij ≥ (cf. thestatements (i) and (iii) of Proposition 6.9). Moreover, by (A.28) we have min k w k w σ ( k ) ∼ . Thus choosing j = σ − ( i ) , so that w i w σ ( j ) = w i , (A.31)yields w i (cid:46) ( M K − ) iσ − ( i ) . This is O (1) by (A.30), and the proof is thus completed. A.5 Hölder continuity of Stieltjes transform
In the proof of Proposition 7.1 we used the following quantitative bound whichstates that the Hölder regularity is preserved under Stieltjes transforms.
Lemma A.7 (Stieltjes transform conserves Hölder regularity) . Let γ ∈ (0 , .Consider an integrable, uniformly γ -Hölder-continuous function ν : R → C , | ν ( τ ) − ν ( τ ) | ≤ C | τ − τ | γ , τ , τ ∈ R , (A.32) where C < ∞ . Then the Stieltjes transform Ξ : H → H of ν , Ξ( ζ ) := (cid:90) R ν ( τ )d ττ − ζ , ζ ∈ H , is also uniformly Hölder continuous with the same Hölder exponent, i.e., | Ξ( ζ ) − Ξ( ζ ) | ≤ C γ (1 − γ ) | ζ − ζ | γ , ζ , ζ ∈ H . (A.33)A similar result can be read off from the estimates of Section 22 of [Mus08].We provide the proof here for the convenience of the reader. Proof.
The L ( R ) -integrability of ν is only needed to guarantee that the Stielt-jes transform is well defined on H . We start by writing Ξ in the form Ξ( ω + i η ) = (cid:90) R ν ( τ ) − ν ( ω ) τ − ω − i η d τ + i π ν ( ω ) , ω ∈ R , η > . (A.34) 144e divide the proof into two steps. First we show that (A.33) holds in thespecial case Im ζ = Im ζ . As the second step we show that (A.33) also holdswhen Re ζ = Re ζ . Together these steps imply (A.33) for general ζ , ζ .Suppose that ζ k = ω k + i η , for some ω , ω ∈ R and η > . Using (A.34) wewrite the difference of the Stieltjes transforms in the form Ξ( ω + i η ) − Ξ( ω + i η ) = i π (cid:2) ν ( ω ) − ν ( ω ) (cid:3) + I + I + I + I , (A.35)where the integrals have been split into the following four parts: I k := ( − k (cid:90) R ν ( τ ) − ν ( ω k ) τ − ω k − i η (cid:110) | τ − ω | ≤ | ω − ω | (cid:111) d τ , k = 1 , .I := ( ν ( ω ) − ν ( ω )) (cid:90) R τ − ω − i η (cid:110) | τ − ω | > | ω − ω | (cid:111) d τ ,I := (cid:90) R ( ν ( τ ) − ν ( ω )) (cid:18) τ − ω − i η − τ − ω − i η (cid:19) (cid:110) | τ − ω | > | ω − ω | (cid:111) d τ . In the regime | τ − ω | > | ω − ω | we have added and subtracted an integral of ν ( ω ) ( τ − ω − i η ) − over τ ∈ R .The first term on the right hand side of (A.35) is less than πC | ω − ω | γ by the hypothesis (A.32). We will show that | I k | ≤ C k | ω − ω | γ , where theconstants C k sum to something less than the corresponding constant on theright hand side of (A.33).Using the γ -Hölder continuity (A.32) of ν , bringing absolute values insidethe integrals, and ignoring η (cid:48) s , it is easy to see that | I | ≤ C γ | ω − ω | γ , and | I | ≤ C γ | ω − ω | γ . (A.36)Due to (A.32), for I we only need to bound the size of the integral. Thereal part of the integral vanishes due to the symmetry. The imaginary part ofthe integral is bounded by (cid:82) R η ( η + λ ) − d λ = π , and thus | I | ≤ πC | ω − ω | γ . (A.37)In order to estimate I we bring absolute values inside the integral, ignore η ’s (cid:12)(cid:12)(cid:12)(cid:12) τ − ω − i η − τ − ω − i η (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ω − ω || τ − ω || τ − ω | , and use the Hölder continuity (A.32) of ν . This yields the first bound below: | I | ≤ C (cid:90) R | ω − ω | (cid:8) | τ − ω | > | ω − ω | (cid:9) | τ − ω || τ − ω − ( ω − ω ) | − γ d τ ≤ C γ (1 − γ ) | ω − ω | γ . (A.38)Plugging this with (A.36) and (A.37) into (A.35) yields | Ξ( ω + i η ) − Ξ( ω + i η ) | ≤ C γ (1 − γ ) | ω − ω | γ . (A.39) 145ow it remains to prove (A.33) in the special case, where ζ k = ω + i η k , forsome ω ∈ R and η , η > . Using again the representation (A.34) we obtain Ξ( ω + i η ) − Ξ( ω + i η ) = (cid:90) R ( ν ( τ ) − ν ( ω )) (cid:18) τ − ω − i η − τ − ω − i η (cid:19) d τ = i (cid:90) R ( η − η ) ( ν ( τ ) − ν ( ω )) d τ ( τ − ω − i η ) ( τ − ω − i η ) . Pulling the absolute values inside the integral yields (cid:12)(cid:12) Ξ( ω + i η ) − Ξ( ω + i η ) (cid:12)(cid:12) ≤ C (cid:90) R | η − η | d τ | τ − ω | − γ √ (cid:0) | τ − ω | + | η − η | (cid:1) ≤ √ C γ (1 − γ ) | η − η | γ . Adding this to (A.39) yields (A.33).
A.6 Cubic roots and associated auxiliary func-tions
Proof of Lemma 9.7 and Lemma 9.15.
Let p k : C → C , k ∈ N , denote anybranch of the inverse of ζ (cid:55)→ ζ k so that p k ( ζ ) k = ζ . We remark that if p k isthe standard complex power function (cf. Definition 9.5) then the conventionalnotation ζ /k is used instead of p k ( ζ ) .The special functions Φ and Φ ± appearing in Lemma 9.6 and Lemma 9.13,respectively, can be stated in terms of the single function Φ( ζ ) := p ( p (1 + ζ ) + ζ ) , (A.40)by rotating ζ and Φ and choosing the functions p and p appropriately. Forexample, if | Re ζ | < , i.e., ζ ∈ (cid:98) C (cf. (9.102)), then Φ( ± i ζ ) = ± iΦ ∓ ( ζ ) ,with the standard definition of the complex powers. In order to treat both thelemmas in the unified way, we hence consider the generic function (A.40) thatis analytic on a simple connected open set D of C such that ± i / ∈ D .Straightforward estimates show that | Φ( ζ ) − Φ( ξ ) | ≤ C | ζ − ξ | / (A.41)and | ∂ ζ Φ( ζ ) | ≤ C (cid:40) | ζ − i | − / + | ζ + i | − / when | ζ | ≤ | ζ | − / when | ζ | > . (A.42)The roots (cid:98) Ω a ( ζ ) defined in both (9.39) and (9.101) are of the form: Ω( ζ ) = α Φ (1) ( ω ζ ) + α Φ (2) ( ω ζ ) . (A.43) 146ere Φ (1) and Φ (2) satisfy (A.40) but with different choices of branches andbranch cuts for the square and the cubic roots. The coefficients α , α , ω , ω ∈ C satisfy | α k | ≤ and | ω k | = 1 for k = 1 , .The perturbation results of Lemma 9.7 and Lemma 9.15 now follow from(A.42) and the mean value theorem: | Φ( ζ + γ ) − Φ( ζ ) | ≤ | γ | sup ≤ ρ ≤ | ∂ ζ Φ( ζ + ργ ) | . (A.44)Indeed, Lemma 9.7 follows directly by choosing D = (cid:8) ζ ∈ C : dist( ζ, G ) ≤ / (cid:9) with G defined in (9.45), and γ := ξ . Since ζ ∈ G ⊂ D the condition (9.46) for c = 1 / guarantees that ζ + ξ ∈ D . As dist( ± i , D ) = 1 / the estimate (9.47)follows using (A.42) in (A.44).In order to prove (9.111) we consider the case ζ = i ( − θ + λ ) and γ = i µ (cid:48) λ ,where θ = ± , | λ − θ | ≥ κ , and | µ (cid:48) | ≤ κ , for some κ ∈ (0 , / . We need tobound the distance between the argument ζ + ργ , of the derivative in (A.44) tothe singular points ± i from below. Assume θ = 1 w.l.o.g. Then the distance of ζ + ργ from − i is bounded from below by (cid:12)(cid:12) ζ + ργ + i (cid:12)(cid:12) ≥ | λ | / , since | ρµ (cid:48) | ≤ κ ≤ / . Similarly, we bound the distance between ζ + ργ and +i from below (cid:12)(cid:12) ζ + ργ − i (cid:12)(cid:12) = (cid:12)(cid:12) ρµ (cid:48) + (1 + ρµ (cid:48) )( λ − (cid:12)(cid:12) ≥ (cid:12)(cid:12) (1 + ρµ (cid:48) )( λ − (cid:12)(cid:12) − ρ | µ (cid:48) |≥ κ + | λ − | / , where for the last estimate we have used the assumption | λ − θ | = | λ − | ≥ κ .These bounds apply for arbitrary ≤ ρ ≤ . Hence they can be applied toestimate the derivative in (A.44) using (A.42). This way we get (cid:12)(cid:12) Φ ( k ) ( ζ + γ ) − Φ ( k ) ( ζ ) (cid:12)(cid:12) ≤ C κ − / min (cid:8) | λ | / , | λ | / (cid:9) | µ (cid:48) | . Applying this in (A.43) yields (9.111).147 ibliography [AEK] Johannes Alt, László Erdős, and Torben Krüger,
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