Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices
BBoundaries of sine kernel universality for Gaussianperturbations of Hermitian matrices
TOM CLAEYS ∗ , THORSTEN NEUSCHEL † , AND MARTIN VENKER ‡ Abstract.
We explore the boundaries of sine kernel universality for the eigenvaluesof Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, westudy for deterministic initial data the time after which Dyson’s Brownian motion ex-hibits sine kernel correlations. We explicitly describe this time span in terms of thelimiting density and rigidity of the initial points. Our main focus lies on cases wherethe initial density vanishes at an interior point of the support. We show that the timeto reach universality becomes larger if the density vanishes faster or if the initial pointsshow less rigidity. Introduction and main results
It is well known that eigenvalues of large random matrices exhibit a highly universalbehavior in the sense that the local limiting distributions depend only on few characteris-tics of the underlying matrix distribution. Typically, symmetries of the random matricesdivide the models into universality classes. In this paper, we will deal with randomHermitian matrices of the form Y n ( t ) := M n + √ tH n , (1.1)where M n is a deterministic n × n Hermitian matrix, t > H n is an n × n randommatrix sampled from the Gaussian Unitary Ensemble (GUE), i.e., from the distributionwith density proportional to e − n Tr( H n ) (1.2)on the space M n of n × n complex Hermitian matrices with respect to the Lebesguemeasure. Equivalently, the entries of H n on and above the diagonal are independent (withindependent real and imaginary parts) with H n,ii , (cid:60) H n,ij , (cid:61) H n,ij , i < j being normallydistributed with mean 0 and variance 1 / ( n (2 − δ ij )).The parameter t will be interpreted as time since √ tH n in (1.1) has the same dis-tribution for fixed t as B t / √ n , where ( B t ) t ≥ is a Brownian motion on the space M n .The division by √ n is convenient for considering eigenvalues in the large n limit as thisrescaling will result in an almost surely compact spectrum. Thus Y n ( t ) has for given t the same distribution as a rescaled Hermitian Brownian motion with initial configuration M n . The corresponding eigenvalue process is called Dyson’s Brownian motion [19], seeFigure 1.Equivalently, Y n ( t ) may be seen as being sampled from a GUE with external source,i.e., a random Hermitian matrix Y n with density proportional to e − n t Tr ( ( Y n − M n ) ) . Key words and phrases.
Random matrices, sine kernel, universality, Dyson’s Brownian motion. a r X i v : . [ m a t h . P R ] D ec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5-2-1.5-1-0.500.511.522.5 t
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5-2-1.5-1-0.500.511.522.5 Figure 1.
Two samples of Dyson’s Brownian motion for t ∈ [0 , n = 20 (left) and n = 100 (right), for M n having equi-spaced eigenvalueson [ − , Y n ( t ) is a random perturbation of the deterministic matrix M n , it is an intriguingquestion to ask how large the time t has to be in order to observe the universal localeigenvalue statistics well-known for many classes of random Hermitian matrices as thesize n tends to infinity. This question has been addressed in many recent papers, see [21]for an overview. From an intuitive point of view, if the eigenvalues of M n are sufficientlydense near a point x ∗ , then the time needed to reach universality is small; near pointswhere the eigenvalues of M n are less dense or near points where there are large spacingsbetween eigenvalues, the time needed to reach universality is larger. We aim at an explicitdescription of the mechanism behind the interaction between the initial configuration ofeigenvalues and the time needed to reach local universality. In particular, we focus onthe case where the limiting eigenvalue density vanishes at an interior point of its support,for which no previous results were available to the best of our knowledge. As usual, wewill measure universality in terms of correlation functions of the eigenvalues.In order to recapitulate the precise terms, let us consider the GUE (1.2) ( M n = 0 and t = 1 in (1.1)) as an example. For 1 ≤ k ≤ n , the k -th correlation function ρ kn of a density P n on R n is the following multiple of the k -th marginal density ρ kn ( x , . . . , x k ) := n !( n − k )! (cid:90) R n − k P n ( x , . . . , x n ) dx k +1 . . . dx n . (1.3)The joint density P n of the eigenvalues of a GUE matrix is proportional to (cid:89) i 2) in the bulk of the spectrum meanslim n →∞ nσ ( x ∗ ) K n (cid:18) x ∗ + unσ ( x ∗ ) , x ∗ + vnσ ( x ∗ ) (cid:19) = sin( π ( u − v )) π ( u − v ) . (1.6)Here, the convergence is locally uniform for u, v ∈ R and the sine kernel at the right-hand side is understood as being 1 if u = v . The limit (1.6) is called local as it concernscorrelations on the scale on which eigenvalues around the point x ∗ have a mean spacingof order 1.The sine kernel has been found to appear for large classes of Hermitian random matrixmodels and is thus called universal. Given the large amount of research on universal-ity of the sine kernel, we can only give a very partial list of references. For the re-sults on bulk universality of the unitary invariant ensembles with density proportional toexp( − n Tr V ( M )) on the matrix space, we mention [18] for the Riemann-Hilbert method,[35, 36] for an approach closer to mathematical physics and [33] for a comparative an-alytic approach. Two recent surveys are [34] and [28]. For bulk universality of Wignermatrices, i.e., random matrices with as many independent entries as possible, we refer to[42] and [20, 11, 21].Apart from these two by now classical situations, the sine kernel has also been shownto appear in a number of different models, among them sparse matrices [25] and matri-ces with correlated entries (but without unitary invariance) [4]. Other classes are moregeneral particle systems with quadratic repulsion [23] or several-matrix-models [22]. Dif-ferent scalings like the unfolding have been considered in [37], giving rates of convergenceand extending the uniformity in statements like (1.6).For the model (1.1), sine kernel universality in the bulk has been shown for fixed t > M n [38, 32] and for t converging to 0 as n → M n is a random matrix [27, 20, 29, 30]. Near the edge of the spectrum, Airykernel universality was shown for a large class of matrices M n in [39] for t > t → M n , it is known that other limitingkernels can appear. If M n has only two distinct eigenvalues, the Pearcey kernel arisesat a critical time [14, 13, 43, 10, 3], and generalizations of the Airy kernel can appearat the edge [1, 2, 7, 8]. Some of these results have been obtained using a representationof the eigenvalue correlation kernel (see (1.9) below) in terms of multiple orthogonalpolynomials. The asymptotic analysis of these multiple orthogonal polynomials can beperformed using Riemann-Hilbert methods if the support of µ n consists of a small numberof points, or in very special situations like equi-spaced points [17], but so far not for generaleigenvalue configurations µ n .Let us now turn to our results about the model (1.1). It is known since works byBr´ezin and Hikami (cf. [13] and references therein) that the eigenvalue distribution P n,t f Y n ( t ) is determinantal in the sense of (1.4) with a kernel also given as a double contourintegral. To write down a formula for the kernel, let the deterministic eigenvalues of M n be a ( n )1 , . . . , a ( n ) n and denote the associated empirical spectral measure by µ n , i.e., µ n := 1 n n (cid:88) j =1 δ a ( n ) j . Moreover let g µ n ( z ) := (cid:90) log( z − x ) dµ n ( x ) , (1.7)where we take the principal branch of the logarithm. Then the correlation functions ρ kn,t of P n,t , defined as in (1.3), satisfy for 1 ≤ k ≤ nρ kn,t ( x , . . . , x k ) = det ( K n,t ( x i , x j )) ≤ i,j ≤ k , (1.8)where the kernel K n,t is defined as K n,t ( x, y ) := e − n t ( x − y )+( x − y ) x nt n (2 πi ) t (cid:90) x + i ∞ x − i ∞ dz (cid:90) γ dw z − w e n t ( ( z − x ) +2 tg µn ( z ) ) e n t (( w − y ) +2 tg µn ( w )) + 1 π ( x − y ) sin (cid:16) ( x − y ) s nt (cid:17) . (1.9)Here, γ is a contour encircling all the points a ( n ) j ’s in the positive sense and such that x + i R and γ intersect precisely at the two points τ = x + is and τ = x − is , with s > τ , τ ) lies in the interior of γ . Explicit contour integralformulas for the correlation kernel K n,t go back to [13], see also [26]. Our formula (1.9)is a variant of these existing formulas which is particularly convenient for asymptoticanalysis in the bulk, as it is decomposed in a way that suggests convergence to the sinekernel. For the convenience of the reader, we give a self-contained proof of (1.8)–(1.9) inAppendix A.Throughout the paper we will make the following assumptions. Assumption 1. The probability measures µ n converge as n → ∞ weakly to an absolutelycontinuous probability measure µ with compact support and with a density which iscontinuous as a function restricted to the support. Assumption 2. There is a compact subset of R which contains the points a ( n ) j for all j and all n .We believe that Assumption 2 is purely technical and could be removed with someextra work.If H n is a GUE matrix, then the limiting measure of √ tH n will be the semicircledistribution dσ t ( x ) := 12 πt √ t − x [ − √ t, √ t ] ( x ) dx. The eigenvalues of the model Y n ( t ) have for any t a global limiting measure µ t in thesense of (1.5). This measure is determined by µ and σ t and is called the additive freeconvolution of µ and σ t [5, 24, 41, 44], in symbols µ t := µ (cid:1) σ t . The measure µ t has a density for any t > ψ t whereas thedensity of µ will be denoted by ψ . As mentioned above, we will focus on bulk correlations. hat is, we want to investigate the local correlations around points in the interior of thesupport of ψ t . We aim at an explicit characterization of bulk points in terms of the initiallimiting measure µ instead of an implicit one in terms of the measure µ t .For that purpose, we define for any x ∈ R the path t (cid:55)→ x t with x t := x + t (cid:90) ( x − u ) ψ ( u ) du ( x − u ) + y t,µ ( x ) , (1.10)with y t,µ given by y t,µ ( x ) := inf (cid:26) y > (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) dµ ( s )( x − s ) + y ≤ t (cid:27) . (1.11)The definitions of this path and of the map x (cid:55)→ y t,µ ( x ) originate in work by Biane [9]about the free additive convolution with a semicircle distribution. It was shown therethat the identity ψ t ( x t ) = y t,µ ( x ) πt (1.12)holds, and this implies in particular that x (cid:55)→ x t defines a bijection between the supportof the function y t,µ and the support of the measure µ t . Note also that the support of y t,µ increases with t and that it contains the support of µ for any t > x ∗ ∈ R , there are two fundamentally different possibilities: either (cid:82) dµ ( s )( s − x ∗ ) isfinite, or it diverges. In the first case, it follows from (1.11) that y t,µ ( x ∗ ) = 0 for t ≤ t cr with the critical time t cr defined as t cr = t cr ( x ∗ ) := (cid:18)(cid:90) dµ ( s )( s − x ∗ ) (cid:19) − . (1.13)In the latter case, we set t cr ( x ∗ ) := 0, and then y t,µ ( x ∗ ) > t > 0. It is easy tosee from (1.10) that t (cid:55)→ x t is a linear path for 0 < t < t cr . From (1.12), we see that ψ t ( x ∗ t ) = 0 for 0 < t ≤ t cr ( x ∗ ) and that ψ t ( x ∗ t ) > t > t cr ( x ∗ ).We note that t cr ( x ∗ ) = 0 if ψ ( x ∗ ) > x ∗ is a zero of ψ of algebraic order0 < κ ≤ 1, i.e., ψ ( x ) ∼ C | x − x ∗ | κ , x → x ∗ , (1.14)where C > ∼ denotes leading order behavior. On the other hand,if x ∗ lies outside the support of µ and also if x ∗ is a zero of ψ of algebraic order κ > 1, wehave t cr ( x ∗ ) > 0. We refer to [15] for more details about the behavior of ψ t in this lastcase.Our first result is merely a slight generalization of existing results, but it constitutesan important natural step for understanding and proving our next results. It deals withfixed times t (independent of n ) bigger than t cr . Theorem 1.1. Let µ n and µ satisfy Assumptions 1 and 2, let x ∗ ∈ R and let t > t cr ( x ∗ ) be fixed. Uniformly for u, v in compacts of R , we have lim n →∞ nψ t ( x ∗ t ) K n,t (cid:18) x ∗ t + unψ t ( x ∗ t ) , x ∗ t + vnψ t ( x ∗ t ) (cid:19) = sin( π ( u − v )) π ( u − v ) , where x ∗ t is given by (1.10) .Remark . The essence of the proof of Theorem 1.1 goes back to the work of Johansson[26] which itself has been inspired by the work of Brezin and Hikami [13]. In [26], M n is a Wigner matrix (independent of H n ), i.e., M n is a random Hermitian matrix withas many independent entries as Hermitian symmetry allows. As the limiting measurefor Wigner matrices with entries having mean 0 and the same variance s > he semicircle distribution σ s and σ s (cid:1) σ t = σ s + t , Johansson’s proof does not deal witha general ψ . It has been subsequently extended to sample covariance matrices [6] andWigner matrices under weak moment assumptions [27]. A variant of Theorem 1.1 thatcovers also real symmetric matrices has been shown in [32]. In that paper, ψ is suchthat ψ t has a connected compact support with strictly positive density on the interior.The case of a general ψ was studied by T. Shcherbina in [38], where she proved bulkuniversality via the method of supersymmetry. Both [32, 38] characterize the bulk pointsas interior of the support of the density ψ t , whereas in Theorem 1.1 the map t (cid:55)→ x ∗ t givesinformation on the origin of the point in terms of ψ and t . Theorem 1.1 also covers caseswhere ψ ( x ∗ ) = 0 but ψ t ( x ∗ t ) > 0. For instance, our result applies also when ψ vanishesat an interior point of its support with exponent κ ≤ 1, or with exponent κ > t > t cr .Theorem 1.1 shows that for Dyson’s Brownian motion the time to local universality inthe bulk is at most O (1). However, Dyson envisioned in [19] that the universal correlations(the “local thermodynamic equilibrium”) should be reached already on time scales slightlybigger than O (1 /n ) for a large class of families of measures µ n . This issue has beenaddressed in more recent years by Erd˝os, Yau and many collaborators, and led to theircelebrated proofs of universality of β - and Wigner ensembles in different symmetry classes(see [21, 29, 20, 30] and references therein). The main condition of Theorem 1.1, namelyweak convergence of µ n to µ , is however not sufficient to have local universality on smalltime-scales, and the time to reach sine kernel universality depends in a more subtle wayon the distribution of the points a ( n ) k . To make this precise, we define quantiles of µ asfollows: we let q ( n ) k ∈ R be such that (cid:90) q ( n ) k −∞ dµ ( s ) = k − n for k = 1 , . . . , n. (1.15)If µ is supported on a single interval, these numbers are uniquely defined since µ isabsolutely continuous. If the support of µ is disconnected, it could happen that some ofthe values q ( n ) k are not uniquely defined. In such cases, we have the freedom to take q ( n ) k to be any value satisfying (1.15).We now define the positive sequence m n /n as the maximal deviation, over k = 1 , . . . , n ,of a point a ( n ) k from its corresponding quantile q ( n ) k , in other words m n := n max ≤ k ≤ n (cid:12)(cid:12)(cid:12) a ( n ) k − q ( n ) k (cid:12)(cid:12)(cid:12) . (1.16)It should be noted that m n in general depends on the particular choice of those quantilesthat are not uniquely defined by (1.15) (in the case that the support is not connected).The sequence m n can be interpreted as a measure for the global rigidity of the eigenvalues a ( n ) k with respect to the measure µ . The simplest example to keep in mind, is the casewhere a ( n ) k = q ( n ) k for 1 ≤ k ≤ n , such that m n = 0. In general, weak convergence of µ n to µ does not imply a bound on the sequence m n , and it can happen that m n grows as n → ∞ . Remark . It is straightforward to verify that (1.16) implies for the Kolmogorov distancebetween µ n and µ that there exists a constant c > 0, depending on µ but not on n , suchthat ˜ m n n := sup x ∈ R | F n ( x ) − F ( x ) | ≤ c ( m n + 1) n , (1.17) here F n and F are the distribution functions of µ n and µ , respectively. Also, for anyinterval I , we have | µ n ( I ) − µ ( I ) | ≤ m n n ≤ c ( m n + 1) n . (1.18)These two facts will be crucial in the proofs of our results.In our next result, we show that sine kernel universality is obtained near x ∗ for times t n which decay slower than ( m n + 1) /n and slower than (log n ) ρ /n for some ρ > 0, if x ∗ is an interior point of the support of µ where its density is positive. Although this isnot surprising in view of recent results in e.g. [20, 29] (see also Remark 1.3 below), webelieve that it is of interest to state and prove this result under explicit conditions on thedistribution µ n via the sequence m n . Theorem 1.2. Let µ n and µ satisfy Assumptions 1 and 2, and let x ∗ belong to the interiorof the support of µ and be such that ψ ( x ∗ ) > . Let t n satisfy t n → , nt n (log n ) ρ → ∞ , and nt n m n +1 → ∞ as n → ∞ , for some ρ > . Uniformly for u, v in compacts of R , we have lim n →∞ nψ t n ( x ∗ t n ) K n,t n (cid:18) x ∗ t n + unψ t n ( x ∗ t n ) , x ∗ t n + vnψ t n ( x ∗ t n ) (cid:19) = sin( π ( u − v )) π ( u − v ) , with x ∗ t n given by (1.10) .Remark . Results of a similar nature have been obtained in the study of bulk uni-versality for Wigner random matrices. In [29], a version of this theorem using vagueconvergence is proved under general but more implicit assumptions on the initial eigen-values. In that paper, the authors assume the so-called local law down to a scale o ( t n ). Asimilar result has also been obtained in [20, Proposition 3.3], using similar saddle pointmethods as ours, but stated under more implicit conditions involving the Stieltjes trans-forms of µ n and µ . Note that Theorem 1.2 slightly improves the lower bound on the timeto universality from O ( n − ε ) for some ε > O ( n − (log n ) ρ ) for some ρ > Remark . Although we defined, for technical reasons, m n in (1.16) as the global max-imal deviation from the quantiles over all eigenvalues a ( n ) k , we believe that especially thedeviations (cid:12)(cid:12)(cid:12) a ( n ) k − q ( n ) k (cid:12)(cid:12)(cid:12) for those quantiles lying close to x ∗ are important.The intuition behind the interplay between the behavior of t n and that of m n as n → ∞ ,is that there might be gaps of mesoscopic size bigger than O ( n − ) between eigenvalues a ( n ) k if m n tends to infinity, and one cannot expect convergence to the sine kernel as longas such mesoscopic gaps are present. The larger such a gap, the longer it will take beforeit is removed by the process. These heuristics will be confirmed and made precise inTheorem 1.4 below.We now focus on the situation of a point x ∗ in the interior of the support of ψ with ψ ( x ∗ ) = 0. We assume that ψ vanishes at x ∗ with exponent κ < 1. Our next theoremshows that for initial configurations that are sufficiently close to quantiles, a time slightlylarger than (cid:0) m n +1 n (cid:1) − κ κ is enough to reach bulk universality, see Figure 2 for an illustration. Theorem 1.3. Let µ n and µ satisfy Assumptions 1 and 2. Let x ∗ ∈ R be a point in theinterior of the support of µ such that (1.14) holds with < κ < . Let t n satisfy t n → , nt κ − κn (log n ) ρ → ∞ , and nt κ − κn m n +1 → ∞ as n → ∞ , for some ρ > . Then locally uniformly in , v lim n →∞ nψ t n ( x ∗ t n ) K n,t n (cid:18) x ∗ t n + unψ t n ( x ∗ t n ) , x ∗ t n + vnψ t n ( x ∗ t n ) (cid:19) = sin( π ( u − v )) π ( u − v ) , with x ∗ t n as in (1.10) . t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5-2-1.5-1-0.500.511.522.5 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5-2-1.5-1-0.500.511.522.5 Figure 2. Two samples of Dyson’s Brownian motion for t ∈ [0 , 1] and n = 100, with M n having eigenvalues which are quantiles of the measures dµ ( x ) = | x | / dx (left) and dµ ( x ) = | x | / dx (right) on [ − , t small, it is clearly visible that the paths near 0 behave differently. Remark . If x ∗ is a point where the density of µ vanishes with exponent κ > 1, wealready mentioned that the density of the free additive convolution µ t = µ (cid:1) σ t has a zerofor t smaller than t cr > 0, see [9] and [15] for a detailed description if κ is an even integer.Then, one cannot expect sine kernel universality to hold for t ≤ t cr , which means thatthe time to reach local universality is drastically bigger than in the case κ < 1. In thethresholding case κ = 1 in (1.14), we expect that the time to reach bulk universality willtend to 0 at a slow rate as n → ∞ , but we do not deal with this particular case in thispaper. Remark . We emphasize that if we replace the conditions nt n m n +1 → ∞ in Theorem1.2 and nt κ − κn m n +1 → ∞ in Theorem 1.3 by the slightly weaker conditions nt n ˜ m n → ∞ and nt κ − κn ˜ m n → ∞ , respectively, then the statements of the theorems remain valid (which followsfrom our proofs). However, we decided to state the theorems in terms of more naturalexplicit conditions on the initial configurations.In our last result, we show that the behavior of the sequence m n is crucial in Theorem1.2 and Theorem 1.3. To that end, we will consider initial configurations µ n which havea gap of size δ n near a point x ∗ , with nδ n → ∞ . We then show that this gap propagatesalong the path (1.10) for times t n which are o ( δ n ). This is illustrated in Figure 3 below. Theorem 1.4. Let µ n = n (cid:80) nk =1 a ( n ) k for some points a ( n ) k , k = 1 , . . . , n , n ∈ N . Let δ n be a sequence converging to in such a way that nδ n → ∞ , as n → ∞ . Suppose that x ∗ ∈ R is such that the intervals [ x ∗ − δ n , x ∗ + δ n ] do not contain any of the starting points ( n )1 , . . . , a ( n ) n . If ε n = o ( δ n ) and if t n = o ( δ n ) , as n → ∞ , we have locally uniformly in u, v , lim n →∞ ε n K n,t n (cid:0) x ∗ t n + ε n u, x ∗ t n + ε n v (cid:1) = 0 . t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5-2-1.5-1-0.500.511.522.5 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x -2.5-2-1.5-1-0.500.511.522.5 Figure 3. Two samples of Dyson’s Brownian motion for t ∈ [0 , n = 100. The initial configurations are obtained by inserting a gap of size0 . 34 near 0 (left) and near 1 / − , t small.As a consequence of the last theorem we present a corollary on gap probabilities. Corollary 1.1. Let x ∗ , δ n , ε n , t n be as in Theorem 1.4. Then lim n →∞ Prob (cid:0) Y n ( t n ) has no eigenvalues in (cid:2) x ∗ t n − ε n , x ∗ t n + ε n (cid:3)(cid:1) = 1 . Proof. It follows from the general theory of determinantal point processes thatProb ( Y n ( t n ) has no eigenvalues in A ) = det(1 − K n,t n | A ) , where det(1 − K n,t n | A ) is the Fredholm determinant associated to the integral operatorwith kernel K n,t n acting on L ( A ). For A = (cid:2) x ∗ t n − ε n , x ∗ t n + ε n (cid:3) , it follows from Theorem1.4 that the operator K n,t n | A converges in trace norm to 0 (cf. [40]), hence the Fredholmdeterminant converges to 1 as n → ∞ . (cid:3) Theorem 1.4 and Corollary 1.1 apply in particular to cases in which the limiting density ψ vanishes at x ∗ with exponent κ > 1. Then the typical distance between consecutiveeigenvalues a ( n ) k close to x ∗ is O ( n − κ +1 ), and the above results imply that each gap of thissize propagates for times t n = o ( n − κ +1 ). In such cases, as explained in Remark 1.5, itcan be expected that sine kernel universality is only reached for times beyond the criticaltime t cr . Near the critical time, non-trivial limiting kernels are expected. We intend tocome back to this in a future publication.The proofs of our results are based on an asymptotic analysis of the double contourintegral representation (1.9) for the kernel K n,t . Since the function n t (( z − x ) + 2 tg µ n ( z ))appearing in the exponential of the double integral depends on µ n and hence on n in arather complicated manner, we will need to carefully study the n -dependence of the saddlepoints and several associated quantities in order to be able to rigorously perform a saddlepoint analysis. On the other hand, to prove convergence to the sine kernel, we will not eed to know the precise leading order behavior of the double integral, and an upperbound will be sufficient. This is a consequence of the convenient form of (1.9), in whichthe sine kernel is explicitly present in the second term.The paper is organized as follows. Section 2 contains the proof of Theorem 1.1. The-orems 1.2, 1.3 and 1.4 are proven in Sections 3, 4 and 5, respectively. We conclude withAppendix A giving a new self-contained proof of the determinantal relations (1.8) for thekernel K n,t given by the double contour integral formula (1.9). Proof of Theorem 1.1 We start by recalling some analytic facts about the free convolution µ (cid:1) σ t . Let us definethe Stieltjes transform of µ by G µ ( z ) := (cid:90) dµ ( x ) z − x for z in the upper half plane C + := { z ∈ C : (cid:61) z > } . Recall the definition of y t,µ in(1.11) for t > 0. Note that (cid:90) dµ ( s )( x − s ) + y = − y (cid:61) G µ ( x + iy ) . The function x (cid:55)→ y t,µ ( x ) is continuous and we writeΩ t,µ = { x + iy ∈ C + | y > y t,µ ( x ) } for the domain above the graph of y t,µ . Biane [9] showed that the function H t,µ ( z ) = z + tG µ ( z ) (2.1)maps the region Ω t,µ conformally to the upper half plane. The graph of y t,µ (the boundaryof Ω t,µ ) is mapped bijectively to the real line. For x real, G µ ( x ) is understood as the limitof G µ ( z ) as z approaches x from the upper half plane. We write F t,µ for the inversefunction of H t,µ , mapping the real line back to the graph of y t,µ .The Stieltjes transform and the density of the free convolution µ t may be recoveredfrom the formulas G µ t ( z ) = G µ ( F t,µ ( z )) and ψ t ( x ) = − π (cid:61) G µ ( F t,µ ( x )) , for x ∈ R . (2.2)Given a reference point x ∗ ∈ R , we define the time evolution x ∗ t = H t,µ ( x ∗ + iy t,µ ( x ∗ )) , (2.3)for points x ∗ on the real line. We have ψ t ( x ∗ t ) = y t,µ ( x ∗ ) πt . It is easy to see that the definition in (2.3) coincides with the one given in (1.10). More-over, using (1.12) it is not difficult to see that with (1.13) we have ψ t ( x ∗ t ) = 0 for any t ≤ t cr and ψ t ( x ∗ t ) > t > t cr . Proof of Theorem 1.1. With the double contour integral formula (1.9) we can write thekernel as 1 c t n K n,t (cid:18) x ∗ t + uc t n , x ∗ t + vc t n (cid:19) = I n ( u, v ) + A n ( u, v ) , ith c t := ψ t ( x ∗ t ) and I n ( u, v ) := e − ntc t ( u − v )+ ctt ( u − v )( x − x ∗ t ) c t (2 πi ) t x + i ∞ (cid:90) x − i ∞ dz (cid:90) γ dw e − f n ( z,w ) z − w , (2.4) A n ( u, v ) = 1 π ( u − v ) sin (cid:18) sc t t ( u − v ) (cid:19) . Here we define f n ( z, w ) = − φ n ( z, u ) + φ n ( w, v ) ,φ n ( z, u ) = n t (cid:34)(cid:18) z − x ∗ t − uc t n (cid:19) + 2 tg µ n ( z ) (cid:35) with g µ n as in (1.7). As in formula (1.9), s > τ inthe upper half plane which is the intersection of γ with the horizontal line x + is , s ∈ R .In order to prove Theorem 1.1, we will show for a suitable choice of the contour γ and thepoint x that A n ( u, v ) converges to the sine kernel as n → ∞ , whereas I n ( u, v ) convergesto 0. We will in particular choose γ and x in such a way that s = πc t t + o (1) , n → ∞ , (2.5)with the o (1) error term uniformly small for u, v in any compact set. Then it is readilyseen that we have lim n →∞ A n ( u, v ) = sin ( π ( u − v )) π ( u − v ) , (2.6)uniformly for u, v in any compact set, if we understand the right hand side as 1 if u = v .The next crucial step is to see that lim n →∞ I n ( u, v ) = 0 (2.7)uniformly in u, v , and this will be done by a two-dimensional saddle point analysis, inwhich delicate estimates are needed, especially because the phase function f n ( z, w ) candepend on n in a complicated manner.Computing the (two-dimensional) complex saddle points of the function f n gives theequations z − x ∗ t − uc t n + tG µ n ( z ) = 0 ,w − x ∗ t − vc t n + tG µ n ( w ) = 0 . Expressing this in terms of the functions H t,µ n ( z ) = z + tG µ n ( z ) gives H t,µ n ( z ) = x ∗ t + uc t n ,H t,µ n ( w ) = x ∗ t + vc t n . Now, as can be seen with help of the definition of y t,µ , the function H t,µ ( z ) is conformalin a neighborhood of x ∗ + iy t,µ ( x ∗ ) = F t,µ ( x ∗ t ), so the inverse F t,µ is conformal in aneighborhood of x ∗ t . Moreover, it follows from the weak convergence of µ n to µ (andeventually using Vitali’s convergence theorem) that H t,µ n converges uniformly to H t,µ in an n -dependent neighborhood U of the point x ∗ + iy t,µ ( x ∗ ) as we stay at a positivedistance from the supports of the measures. So there exists an index N such that for all n > N the functions H t,µ n are conformal mappings from U onto a neighborhood of x ∗ t .Hence, for large n , we can consider the inverse functions F t,µ n of H t,µ n in a neighborhood f x ∗ t that is independent of n , so that we obtain a specific pair of solutions for the saddlepoint equations by z n = x n + iy n = F t,µ n (cid:18) x ∗ t + uc t n (cid:19) ,w n = F t,µ n (cid:18) x ∗ t + vc t n (cid:19) and the corresponding complex conjugate solutions. In total we find four two-dimensionalsaddle points ( z n , w n ) , ( z n , w n ) , ( z n , w n ) and ( z n , w n ). In (2.4) we now choose specificcontours of integration. The integral in the z -plane is taken along the vertical line through x = x n = (cid:60) F t,µ n (cid:16) x ∗ t + uc t n (cid:17) . The parts of this line in the lower and in the upper partof the complex plane give paths of descent for the phase function φ n ( z, u ). For instance,for τ > ddτ (cid:60) φ n ( x n + iτ, u ) = nτt (cid:20) t (cid:90) dµ n ( a )( x n − a ) + τ − (cid:21) , where (cid:82) dµ n ( a )( x n − a ) + τ is strictly decreasing in τ , and the right hand side of the above equationvanishes if and only if τ = y n = (cid:61) F t,µ n (cid:16) x ∗ t + uc t n (cid:17) . Moreover, we have β n := − d dτ (cid:12)(cid:12)(cid:12) τ = y n (cid:60) φ n ( x n + iτ, u ) = 2 ny n (cid:90) dµ n ( a )(( x n − a ) + y n ) > , (2.8)so that z n is a simple saddle point for φ n ( z, u ). It is clear by the same argumentationthat w n is a simple saddle point for φ n ( w, v ). For the integral in the w -plane we constructa path γ = γ n using the graph of the function y t,µ n . Since the supports of the measures µ n are all contained in a compact set independent of n , we can find a finite intervalindependent of n (but depending on t ), say J , containing all the supports of the functions y t,µ n . Indeed, this follows easily from the fact that y t,µ n ( x ) = 0 for dist( x, supp( µ n )) ≥ √ t ,which is a direct consequence of the definition (1.11). The path γ n starts at a real pointto the right of J and follows the graph of y t,µ n , passes the saddle points z n and w n on theway, and finally returns for a last time back to the real axis at a point located to the leftof J . We complete the path γ n just by going back using the complex conjugate path inthe lower half plane. This establishes a path of descent for the phase function − φ n ( w, v )of the integral in the w -plane passing through the saddle points w n , w n . We can verifythis, for instance, by parametrizing the upper part of γ n by ˜ γ n ( τ ) = τ + iy t,µ n ( τ ) (usingthe opposite orientation for convenience) and computing ddτ (cid:60) φ n (˜ γ n ( τ ) , v ) = nt (cid:18) H t,µ n (˜ γ n ( τ )) − x ∗ t − vc t n (cid:19) . With these choices we get contours passing though each of the four saddle points alongpaths of descent. Now we will show that˜ I n ( u, v ) = c t (2 πi ) te ntc t ( u − v ) − ctt ( u − v )( x − x ∗ t ) I n ( u, v )= x n + i ∞ (cid:90) x n − i ∞ dz (cid:90) γ n dw e − f n ( z,w ) z − w = x n + i ∞ (cid:90) x n − i ∞ dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w converges to zero as n → ∞ . We remark in passing that we excluded the “gauge factor”exp( − ntc t ( u − v ) + c t t ( u − v )( x − x ∗ t )) from I n for notational convenience only. Toshow the vanishing of ˜ I n ( u, v ), we split the integration contours in several parts. As an be expected, the main contribution to the double integral will come from smallneighbourhoods of the saddle points. For technical reasons, we define L n = log n √ n , (2.9)and split the integral into seven parts ˜ I n ( u, v ) = (cid:80) k =1 ˜ I ( k ) n in the following way:˜ I (1) n = z n + i ∞ (cid:90) z n + iL n dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w , ˜ I (2) n = z n − iL n (cid:90) z n − i ∞ dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w , ˜ I (3) n = z n − iL n (cid:90) z n + iL n dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w , ˜ I (4) n = z n + iL n (cid:90) z n − iL n dz (cid:90) γ n ∩ U Ln ( w n ) dw e φ n ( z,u ) − φ n ( w,v ) z − w , where U L n ( w n ) denotes a disk of radius L n centered at w n ,˜ I (5) n = z n + iL n (cid:90) z n − iL n dz (cid:90) γ n \ U Ln ( w n ) dw e φ n ( z,u ) − φ n ( w,v ) z − w , ˜ I (6) n = z n + iL n (cid:90) z n − iL n dz (cid:90) γ n ∩ U Ln ( w n ) dw e φ n ( z,u ) − φ n ( w,v ) z − w , and ˜ I (7) n = z n + iL n (cid:90) z n − iL n dz (cid:90) γ n \ U Ln ( w n ) dw e φ n ( z,u ) − φ n ( w,v ) z − w . To estimate the above integrals, we will need a rough bound for the length of the curve γ n , which is obtained in the following lemma. We will prove a more general version as itwill be needed later on, however here we only apply it for the case of a fixed t . Lemma 2.1. Let L ( γ n ) denote the length of γ n . Then for any positive bounded sequence t n we have L ( γ n ) = O (cid:0) n √ t n (cid:1) , n → ∞ . Proof. We assume that the intervals of positivity of the function y t n ,µ n are contained insidethe bounded interval J . In order to estimate the length of γ n , which is constructed fromthe the graph of y t n ,µ n as described above, it is necessary to estimate the length on theseintervals of positivity which form the support of y t n ,µ n . We aim to find an upper boundfor the number of intervals of monotonicity into which the support can be partitioned. Inorder to do so we will find an estimate for the number of vanishing points of the derivativeof y t n ,µ n inside its support. If I is an interval on which y t n ,µ n is monotonically increasingor decreasing, we have that the length of its graph on I is bounded by L ( I ) √ t n , where ( I ) denotes the length of I and we use the fact that y t n ,µ n ( x ) ≤ √ t n for all x ∈ R . Onthe intervals of positivity we have the equality n (cid:88) k =1 x − a ( k ) n ) + y t n ,µ n ( x ) = nt n , which we can write as n (cid:89) j =1 (cid:0) y t n ,µ n ( x ) + ( x − a ( j ) n ) (cid:1) = t n n n (cid:88) k =1 (cid:89) j (cid:54) = k (cid:0) y t n ,µ n ( x ) + ( x − a ( j ) n ) (cid:1) . Hence, the function w = w n = y t n ,µ n satisfies the algebraic equation n (cid:89) j =1 (cid:0) w ( x ) + ( x − a ( j ) n ) (cid:1) = t n n n (cid:88) k =1 (cid:89) j (cid:54) = k (cid:0) w ( x ) + ( x − a ( j ) n ) (cid:1) . (2.10)As the derivatives of y t n ,µ n and w vanish at the same points inside the support, it issufficient to find an upper bound for the points of vanishing of the derivative of thealgebraic function w defined by (2.10) on its entire associated Riemann surface. To thisend, we rewrite equation (2.10) in the form Q n ( x, w ) = x n + n − (cid:88) k =0 p k,n ( w ) x k = 0 , (2.11)where p k,n ( w ) are polynomials in w of degree at most n . Let us assume that the derivativeof w vanishes at some point x of its Riemann surface. Then by differentiation we seethat the pair ( x , w ( x )) satisfies the equations Q n ( x , w ( x )) = 0 , ∂Q n ∂x ( x , w ( x )) = 0 . Hence, at the point w = w ( x ) the (univariate) polynomial Q n ( x, w ( x )) has a multipleroot at x = x . This means that w ( x ) is a root of the corresponding discriminantwhich, in view of (2.11), is of degree at most 2 n (2 n − { w , . . . , w n (2 n − } of at most 2 n (2 n − 1) different values which w ( x ) can take.But for every possible value w ( x ) = w k there are at most 2 n different solutions of Q n ( x, w ( x )) = 0 considered as a polynomial equation in x . From this we obtain a set { x , . . . , x n (2 n − } of at most 4 n (2 n − 1) different values that x can take, which meansthat there are at most 4 n (2 n − 1) points x such that w (cid:48) ( x ) = 0. Hence, on the intervalsof positivity of the function y t n ,µ n its derivative can vanish at at most 4 n (2 n − 1) differentpoints. But from this we infer that throughout the support of y t n ,µ n , there are at most4 n (2 n − 1) + 1 many intervals of monotonicity. This gives for the length of γ n L ( γ n ) = O (cid:0) n √ t n (cid:1) , as n → ∞ . (cid:3) In what follows, all estimates hold for sufficiently large values of n , and we will useconstants η, ˜ η, ˆ η > n and also independent of u, v (for u, v inany compact set), but which can change their values at different occasions without beingmentioned explicitly. Also, constants implied by asymptotic O ( · ) or o ( · ) notations as n → ∞ can be chosen independent of u, v for u, v in any compact set. Moreover, althoughterms of the form √ t could be absorbed by constants we will write them explicitly in favorof later references. stimation of ˜ I (1) n . First, we observe that for ( z, w ) on [ z n + iL n , z n + i ∞ ] × γ n we have d − n := 1min | z − w | ≤ ηL n . (2.12)Moreover, on γ n the function − φ n ( w, v ) takes its maximum in w = w n , so that by Lemma2.1 we obtain | ˜ I (1) n | ≤ ˜ ηn √ td n e (cid:60){ φ n ( z n ,u ) − φ n ( w n ,v ) } z n + i ∞ (cid:90) z n + iL n | dz | e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } . A computation using the definition of the saddle points shows z n − w n = O (cid:18) n (cid:19) , (2.13)and also − f n ( z n , w n ) = φ n ( z n , u ) − φ n ( w n , v ) = O (1) , (2.14)as n → ∞ . This gives | ˜ I (1) n | ≤ ˜ ηn √ td n z n + i ∞ (cid:90) z n + iL n | dz | e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } ≤ ˜ ηn √ td n e (cid:60){ φ n ( z n + iL n ,u ) − φ n ( z n ,u ) } z n + i ∞ (cid:90) z n + iL n | dz | e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } . A complex Taylor expansion for φ n ( z, u ) around z n yields φ n ( z, u ) = φ n ( z n , u ) + 12 φ (cid:48)(cid:48) n ( z n , u )( z − z n ) + R ( z )with | R ( z ) | ≤ max | s − z n | = r | φ n ( s, u ) | r | z − z n | − | z − z n | r ≤ n ˆ η | z − z n | ≤ n ˆ ηL n , (2.15)for suitably small r > | z − z n | ≤ L n . By (2.8) and by the fact that nL n → n → ∞ , we get˜ ηn √ td n e (cid:60){ φ n ( z n + iL n ,u ) − φ n ( z n ,u ) } = O (cid:18) n √ td n e − βn L n (cid:19) , n → ∞ . Finally, we have z n + i ∞ (cid:90) z n + iL n | dz | e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } ≤ z n + i ∞ (cid:90) z n + iL n | dz | e (cid:60) (cid:26) − (cid:16) z n − x ∗ t − uctn (cid:17) − tg µn ( z n )+ (cid:16) z − x ∗ t − uctn (cid:17) +2 tg µn ( z ) (cid:27) , and the right-hand side converges as n → ∞ to F t,µ ( x ∗ t )+ i ∞ (cid:90) F t,µ ( x ∗ t ) | dz | e (cid:60) { − ( F t,µ ( x ∗ t ) − x ∗ t ) − tg µ ( F t,µ ( x ∗ t ))+( z − x ∗ t ) +2 tg µ ( z ) } < ∞ . Together this gives by (2.9), (2.12) and the fact that lim n →∞ β n /n exists and is positive,˜ I (1) n = O (cid:18) n √ td n e − βn L n (cid:19) = O (cid:18) n / √ t log n e − η (log n ) (cid:19) , n → ∞ , or some constant η > I (2) n and ˜ I (3) n can be treated in a very similar fashion, where the estimationof ˜ I (3) n is slightly simpler as the contours of integration stay bounded. Estimation of ˜ I (4) n . Recall that H t,µ n maps the part of γ n in the upper half plane bijec-tively and conformally to a part of the real line. This implies the identityarctan y (cid:48) t,µ n ( (cid:60) w ) = − arg H (cid:48) t,µ n ( w ) , w ∈ γ n ∩ U L n ( w n ) . (2.16)As n → ∞ , arg H (cid:48) t,µ n ( w ) → arg H (cid:48) t,µ ( x ∗ ) ∈ ( − π/ , π/ y (cid:48) t,µ n ( (cid:60) w )remains bounded for large n , and that the length of the contour γ n ∩ U L n ( w n ) is O ( L n ) as n → ∞ . Hence, we can use an arc-length parametrization γ n : [ − (cid:96) n , ˜ (cid:96) n ] → γ n ∩ U L n ( w n ),with γ n (0) = z n and (cid:96) n , ˜ (cid:96) n ≤ ηL n , to compute the integral ˜ I (4) n . This easily leads to theupper bound | ˜ I (4) n | ≤ e max (cid:60) ( φ n ( z,u ) − φ n ( w,v )) z n + iL n (cid:90) z n − iL n | dz | (cid:90) ˜ (cid:96) n − (cid:96) n | ds | | z − γ n ( s ) | , (2.17)where the maximum is taken over ( z, w ) ∈ [ z n − iL n , z n + iL n ] × ( γ n ∩ U L n ( w n )). If wewrite γ n ( L n σ ) = z n + L n ˜ γ n ( σ ), s = L n σ , and z = z n + iL n ζ , then | ˜ I (4) n | ≤ L n e max (cid:60) ( φ n ( z,u ) − φ n ( w,v )) 1 (cid:90) − | dζ | (cid:90) η − η | dσ | | iζ − ˜ γ n ( σ ) | . (2.18)The remaining integral remains bounded by the dominated convergence theorem andsince the angle between γ n and z n + i R cannot become small. By construction ofthe integration contours as descent paths, we also have max (cid:60) ( φ n ( z, u ) − φ n ( w, v )) = (cid:60) ( φ n ( z n , u ) − φ n ( w n , v )) and it follows that˜ I (4) n = O ( L n ) , n → ∞ . (2.19)By symmetry, the integral ˜ I (6) n can be estimated by the same arguments, so it remainsto treat ˜ I (5) n and ˜ I (7) n , which again are of the same type so we only have to deal with ˜ I (5) n . Estimation of ˜ I (5) n . On the contour [ z n − iL n , z n + iL n ] × ( γ n \ U L n ( w n )) we have˜ d − n := 1min | z − w | ≤ ηL n , (2.20)hence the double integral can be estimated by decoupled integrals, | ˜ I (5) n | ≤ d n z n + iL n (cid:90) z n − iL n | dz | e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } (cid:90) γ n \ U Ln ( w n ) | dw | e −(cid:60){ φ n ( w,v ) − φ n ( w n ,v ) } . By a Taylor expansion of φ n ( z, u ) around z n and arguments already used above we have z n + iL n (cid:90) z n − iL n | dz | e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } = O (cid:18) √ n (cid:19) , n → ∞ . To estimate the remaining integral we define { w + n , w − n } = γ n ∩ ∂U L n ( w n ). By Lemma 2.1,we obtain as n → ∞ , | ˜ I (5) n | = O (cid:18) n √ t ˜ d n √ n (cid:19) (cid:110) e −(cid:60) { φ n ( w + n ,v ) − φ n ( w n ,v ) } + e −(cid:60) { φ n ( w − n ,v ) − φ n ( w n ,v ) } (cid:111) . e have by one last Taylor expansion, similarly as before, since (cid:60) φ (cid:48)(cid:48) n ( w n , v ) /n tends to apositive constant, (cid:60) (cid:8) φ n ( w + n , v ) − φ n ( w n , v ) (cid:9) ≥ ηnL n (2.21)and an analogous estimate holds for (cid:60) { φ n ( w − n , v ) − φ n ( w n , v ) } . This gives | ˜ I (5) n | = O (cid:18) n √ t ˜ d n √ n e − ηnL n (cid:19) = O (cid:32) n √ te − η (log n ) log n (cid:33) , n → ∞ . (2.22)Collecting all these estimates on ˜ I n ( u, v ), we obtain (2.7). By taking into account that s = (cid:61) z n = (cid:61) F t,µ n (cid:18) x ∗ t + uc t n (cid:19) → (cid:61) F t,µ ( x ∗ t ) = − πψ t ( x ∗ t ) t, we also obtain (2.6) via (2.5), and this completes the proof of the theorem. (cid:3) Proof of Theorem 1.2 For the proof of Theorem 1.2, we follow the same strategy as for the proof of Theorem1.1 in Section 2, but there are quite some technical issues to be dealt with differently,because of the fact that t = t n → 0. The most essential difference is that the saddlepoints z n and w n approach the real line as n → ∞ . We need to control the speed ofconvergence in order to obtain a saddle point approximation. Auxiliary results Lemma 3.1. Let x ∗ be an interior point of the support of µ , and suppose that δ > issuch that the interval [ x ∗ − δ, x ∗ + δ ] belongs to the interior of the support. Let m n bedefined by (1.16) . Then there exists a constant C > and an index N ∈ N such that forall n > N , ε > and x ∈ [ x ∗ − δ, x ∗ + δ ] we have the inequality | G µ n ( x + iε ) − G µ ( x + iε ) | ≤ ˜ m n πnε ≤ C ( m n + 1) nε . Proof. Let F n and F denote the distribution functions of µ n and µ , respectively. Wedefine g ( s ) := 1 x + iε − s and choose a < b such that the supports of µ and µ n lie in the interior of [ a, b ]. Then byintegration by parts and the absolute continuity of g we have G µ n ( x + iε ) − G µ ( x + iε ) = (cid:90) ba g d ( F n − F ) = − (cid:90) ba ( F n − F ) dg = − (cid:90) ba ( F n − F )( s ) g (cid:48) ( s ) ds. By (1.17), we have sup s ∈ [ a,b ] | F n ( s ) − F ( s ) | ≤ ˜ m n n ≤ c ( m n +1) n , and hence | G µ n ( x + iε ) − G µ ( x + iε ) | ≤ ˜ m n n (cid:90) ba x − s ) + ε ds = ˜ m n nε (cid:18) arctan (cid:18) b − xε (cid:19) − arctan (cid:18) a − xε (cid:19)(cid:19) ≤ ˜ m n πnε ≤ C ( m n + 1) nε . In the following, integral expressions of the form (cid:82) βα f ( s ) dµ n ( s ) are always to be un-derstood as (cid:82) [ α,β ] f ( s ) dµ n ( s ). Lemma 3.2. Under the conditions of Theorem 1.2, for any ε > , there exist δ > and n > such that y t n ,µ n ( x ) ≥ (1 − ε ) t n ψ ( x ) , for any n > n , x ∈ [ x ∗ − δ, x ∗ + δ ] .Proof. For any α n > x ∈ R , we have the inequality1 n n (cid:88) k =1 α n + ( a ( n ) k − x ) ≥ α n (cid:90) x + α n x − α n dµ n ( s ) . We define α n = (1 − ε ) ψ ( x ∗ ) t n , so that by (1.18) we have (cid:90) x + α n x − α n dµ n ( s ) ≥ (cid:90) x + α n x − α n dµ ( s ) − m n n , and hence 1 n n (cid:88) k =1 α n + ( a ( n ) k − x ) ≥ α n (cid:18)(cid:90) x + α n x − α n dµ ( s ) − m n n (cid:19) . If x is sufficiently close to x ∗ and α n → n → ∞ , there exists n such that for n > n we have (cid:90) x + α n x − α n dµ ( s ) ≥ (cid:16) − ε (cid:17) (cid:90) x ∗ + α n x ∗ − α n dµ ( s ) ≥ (cid:16) − ε (cid:17) ψ ( x ∗ ) α n . We then have1 n n (cid:88) k =1 α n + ( a ( n ) k − x ) ≥ ψ ( x ∗ ) (cid:16) − ε (cid:17) α − n − ˜ m n nα n ≥ (1 − ε ) ψ ( x ∗ ) α − n = 1 t n , if nα n / ˜ m n → ∞ , which is true since nt n / ( m n + 1) → ∞ as n → ∞ . By definition of y t n ,µ n , we have that α n ≤ y t n ,µ n ( x ), and this proves the result. (cid:3) Lemma 3.3. Under the conditions of Theorem 1.2, we have lim n →∞ y t n ,µ n ( x ) πt n = ψ ( x ) , uniformly for x ∈ [ x ∗ − δ, x ∗ + δ ] with δ sufficiently small.Proof. Combining (2.2) and (1.12), we obtain y t n ,µ n ( x ) πt n = − π (cid:61) G µ n ( x + iy t n ,µ n ( x ))for real x . Using Lemma 3.2 and Lemma 3.1, we can conclude that y t n ,µ n ( x ) πt n = − π (cid:61) G µ ( x + iy t n ,µ n ( x )) + O (cid:18) ˜ m n nt n (cid:19) → ψ ( x )as n → ∞ with nt n / ˜ m n → ∞ , uniformly for x sufficiently close to x ∗ . (cid:3) emma 3.4. Under the conditions of Theorem 1.2, for u, v in a compact set, there existsa constant M > such that z n = z n ( u, t n ) and w n = w n ( v, t n ) satisfy | z n − w n | ≤ Mn , for sufficiently large n .Proof. We have z n = F t n ,µ n (cid:18) x ∗ t n + uc t n n (cid:19) , w n = F t n ,µ n (cid:18) x ∗ t n + vc t n n (cid:19) . To prove the result, it is sufficient to show that | F (cid:48) t n ,µ n ( x ) | remains uniformly boundedfor x ∈ R and | x − x ∗ t n | < Kn for a sufficiently large constant K > 0. We have F (cid:48) t n ,µ n ( x ) = 1 H (cid:48) t n ,µ n ( F t n ,µ n ( x )) . Using the fact that (cid:82) dµ n ( s ) | z − s | = t n for z = F t n ,µ n ( x ) on positive parts of the graph of y t n ,µ n ,we have | H (cid:48) t n ,µ n ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12) − t n (cid:90) dµ n ( s )( z − s ) (cid:12)(cid:12)(cid:12)(cid:12) (3.1)= t n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dµ n ( s ) | z − s | − (cid:90) dµ n ( s )( z − s ) (cid:12)(cid:12)(cid:12)(cid:12) = t n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) | z − s | − ( z − s ) | z − s | dµ n ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . The modulus of the latter integral can be bounded below by the absolute value of its realpart, and this gives | H (cid:48) t n ,µ n ( z ) | ≥ t n ( (cid:61) z ) (cid:90) dµ n ( s ) | z − s | ≥ t n ( (cid:61) z ) (cid:90) (cid:60) z n + t n (cid:60) z n − t n dµ n ( s ) | z − s | . By Lemma 3.2, this can be further estimated by | H (cid:48) t n ,µ n ( z ) | ≥ Ct − n (cid:90) (cid:60) z n + t n (cid:60) z n − t n dµ n ( s ) , for some ˜ C > 0, and the right hand side of the latter expression is bounded below bya positive constant since (cid:82) (cid:60) z n + t n (cid:60) z n − t n dµ n ( s ) ∼ ψ ( x ∗ ) t n as n → ∞ , which follows from astraightforward argument using (1.18) and the fact that nt n / ˜ m n → ∞ . (cid:3) Asymptotics of A n It is now straightforward to give the asymptotics of A n . We have A n ( u, v ) = 1 π ( u − v ) sin (cid:18) sc t n t n ( u − v ) (cid:19) , where s is the imaginary part of the intersection of γ with the vertical line through x = x n = (cid:60) F t n ,µ n (cid:16) x ∗ t n + uc tn n (cid:17) . In other words, if we make the same choice of integrationcontours in (2.4) as before, then s = (cid:61) F t n ,µ n (cid:18) x ∗ t n + uc t n n (cid:19) = y t n ,µ n ( x n ) . ince x n → x ∗ , c t n → ψ ( x ∗ ) as n → ∞ and t n → 0, it follows from Lemma 3.3 that sc tn t n → π , and consequently lim n →∞ A n ( u, v ) = sin( π ( u − v )) π ( u − v ) . Estimation of I n Here, we follow the estimates done in Section 2 for the proof of Theorem 1.1. As alreadyspecified in the analysis of A n , we take the same integration contours in (2.4) as before, byfollowing the graph of y t n ,µ n and its complex conjugate for the contour γ n , and by taking x = x n = (cid:60) F t n ,µ n ( x ∗ t n + uc tn n ). At first sight, the fact that the saddle points z n and w n ofthe phase function φ n approach the real line may appear problematic, as one may expectthat the contributing neighborhoods of z n , z n and w n , w n will overlap. However, this isnot the case, essentially because the second derivative of the phase function φ n blowsup rapidly, which means that the contributing neighborhoods to the integral I n becomesmall as well. More concretely, in view of (2.8), we observe that β n = − d dτ (cid:12)(cid:12)(cid:12) τ = y n (cid:60) φ n ( x n + iτ, u ) = 2 ny n (cid:90) dµ n ( a )(( x n − a ) + y n ) ≥ ny n t n + y n ) (cid:90) x n + t n x n − t n dµ n ( s ) . Using Lemma 3.3 and the fact that n (cid:82) x n + t n x n − t n dµ n ( s ) > εnt n for sufficiently small ε > nt n / ( m n + 1) → ∞ ), we obtain for some K > β n ≥ Knt n . With this in mind, we can proceed along the same lines as in the proof of Theorem 1.1,splitting the integral ˜ I n in seven parts ˜ I ( n ) n , . . . , ˜ I (7) n . Since the imaginary parts of thesaddle points are proportional to t n as n → ∞ , we need to take L n such that L n /t n → L n = (cid:114) t n n (log n ) ρ . Estimation of ˜ I (4) n . The estimates (2.17), (2.18), and (2.19) remain valid using the samecalculations as in Section 2, provided that we can show that the length of γ n ∩ U L n ( w n )is O ( L n ) as n → ∞ , and that the angle between γ n and z n + i R stays away from 0 forlarge n . To see this, note that for w ∈ γ n ∩ U L n ( w n ), H (cid:48) t n ,µ n ( w ) = 1 − t n (cid:90) dµ n ( s )( w − s ) = t n (cid:18)(cid:90) dµ n ( s ) | w − s | − (cid:90) dµ n ( s )( w − s ) (cid:19) = t n (cid:90) | w − s | − ( w − s ) | w − s | dµ n ( s ) . ence, (cid:60) H (cid:48) t n ,µ n ( w ) = 2 ty t n ,µ n ( (cid:60) w ) (cid:90) dµ n ( s ) | w − s | ≥ t n y t n ,µ n ( (cid:60) w ) (cid:90) (cid:60) w + y tn,µn ( (cid:60) w ) (cid:60) w − y tn,µn ( (cid:60) w ) dµ n ( s ) | w − s | ≥ η t n y t n ,µ n ( (cid:60) w ) (cid:90) (cid:60) w + y tn,µn ( (cid:60) w ) (cid:60) w − y tn,µn ( (cid:60) w ) dµ n ( s ) . By Lemma 3.3, (1.18), and the fact that nt n / ( m n + 1) → ∞ , this is bounded below bya positive constant. It is also easily seen that | H (cid:48) t n ,µ n ( w ) | ≤ 2, and it follows that theargument of H (cid:48) t n ,µ n ( w ) remains bounded away from ± π/ 2. By (2.16), this implies that y (cid:48) t n ,µ n ( (cid:60) w ) is bounded for w ∈ γ n ∩ U L n ( w n ), and then it easily follows that the length of γ n ∩ U L n ( w n ) is O ( L n ) as n → ∞ , and that the angle between γ n and z n + i R does notapproach 0. Estimation of I (1) n . Using the fact that y (cid:48) t n ,µ n is bounded near x ∗ (which we showed inthe above paragraph), we see easily that the inequality (2.12), where the minimum isover ( z, w ) ∈ [ z n + iL n , z n + i ∞ ) × γ n , still holds.By Lemma 3.4, we still have (2.13). Using the fact that n | g µ n ( z n ) − g µ n ( w n ) | = n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) z n w n G µ n ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:90) z n w n | G µ n ( s ) | | ds | , and combining this with Lemma 3.2 and Lemma 3.1, we obtain that (2.14) also holds.In the remaining part of the analysis of I (1) n , we follow similar estimates as in the proofof Theorem 1.1, but we need to estimate the error term R ( z ) in the Taylor expansion ina different way. Instead of (2.15), we have for | z − z n | ≤ L n | R ( z ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) zz n dξ (cid:90) ξ z n dξ (cid:90) ξ z n dξ φ (cid:48)(cid:48)(cid:48) n ( ξ ; u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ max | φ (cid:48)(cid:48)(cid:48) n ( ξ ; u ) | L n , (3.2)where we integrate over line segments, and where the maximum is over the line segmentbetween z n and z . Next, given ξ on this line segment, we let γ be the circle of radius (cid:61) ξ ≥ ( (cid:61) z n − L n ) around ξ , and we use Cauchy’s theorem to estimate the maximum,max | φ (cid:48)(cid:48)(cid:48) n ( ξ ; u ) | = (cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) γ φ (cid:48)(cid:48) n ( ζ ; u )( ζ − ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:61) ξ max ζ ∈ γ | φ (cid:48)(cid:48) n ( ζ ; u ) | ≤ n max ζ ∈ γ | t n G (cid:48) µ n ( ζ ) | t n ( (cid:61) z n − L n ) . (3.3)By Lemma 3.2, it follows that (cid:61) z n is bounded below by ˜ ct n for a sufficiently small constant˜ c > 0. Since L n = o ( t n ) as n → ∞ , we have (cid:61) z n − L n ≥ ˜ c t n , (cid:61) ζ ≥ (cid:61) ξ ≥ 12 ( (cid:61) z n − L n ) ≥ ˜ c t n , for n sufficiently large. Now we can use Lemma 3.3 to conclude that there exists asufficiently small constant (cid:98) c > 0, uniform in ζ , such that y (cid:98) ct n ,µ n ( (cid:60) ζ ) ≤ ˜ c t n ≤ (cid:61) ζ . Butfor such a constant (cid:98) c , we have (cid:12)(cid:12) G (cid:48) µ n ( ζ ) (cid:12)(cid:12) ≤ (cid:90) | ζ − s | dµ n ( s ) ≤ (cid:98) ct n , by definition of y t n ,µ n . Substituting this in (3.2) and (3.3), we obtain | R ( z ) | ≤ nL n ˜ ct n (cid:18) (cid:98) c (cid:19) = o ( β n L n ) , n → ∞ . hen, proceeding as in the proof of Theorem 1.1, we get˜ I (1) n = O (cid:18) n √ t n d n e − βn L n (cid:19) = O (cid:16) n e − η (log n ) ρ (cid:17) = o (1) , n → ∞ . The same bound applies to ˜ I (2) n and ˜ I (3) n . Estimation of ˜ I (5) n . For ˜ I (5) n and ˜ I (7) n , we first note that (2.20) still holds. Using a Taylorexpansion and error estimate similar to the one for ˜ I (1) n , one verifies that (2.21) improvesto (cid:60) (cid:8) φ n ( w + n , v ) − φ n ( w n , v ) (cid:9) ≥ η nt n L n because t n n (cid:60) φ (cid:48)(cid:48) n ( w n , v ) is bounded in absolute value und bounded away from zero as n →∞ . This then leads to, instead of (2.22), | ˜ I (5) n | = O (cid:18) n t n ˜ d n √ n e − η ntn L n (cid:19) = O (cid:16) n e − η (log n ) ρ (cid:17) , n → ∞ . Combining the above estimates, we get that lim n →∞ ˜ I n = 0, which completes the proofof Theorem 1.2.It remains to consider the factor exp( − nt n c tn ( u − v ) + c tn t n ( u − v )( x − x ∗ t n )) that wasneglected when passing from I n to ˜ I n . We will show that x − x ∗ t n = x n − x ∗ t n = O ( t n ),which clearly suffices. Expanding F t n ,µ n around H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )), we get by theconvergence of c t n and the uniform boundedness of F (cid:48) t n ,µ n around x ∗ t n (see the proof ofLemma 3.4) F t n ,µ n (cid:18) x ∗ t n + uc t n n (cid:19) = x ∗ + O (cid:18) n + | H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )) − H t n ,µ ( x ∗ + iy t n ,µ ( x ∗ )) | (cid:19) . Equation (3.1) and t n (cid:82) dµ n ( s ) | z − s | = 1 show that H (cid:48) t n ,µ n is uniformly bounded. Since byLemma 3.3 y t n ,µ n ( x ∗ ) − y t n ,µ ( x ∗ ) = O ( t n ), we have H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )) − H t n ,µ n ( x ∗ + iy t n ,µ ( x ∗ )) = O ( t n ). Furthermore, Lemmas 3.1 and 3.3 show that H t n ,µ n ( x ∗ + iy t n ,µ ( x ∗ )) − H t n ,µ ( x ∗ + iy t n ,µ ( x ∗ )) = O (1 /n ). It remains to bound | x ∗ − x ∗ t n | . Since H t n ,µ maps x ∗ + iy t n ,µ ( x ∗ ) to the real line, we have x ∗ − x ∗ t n = t n (cid:60) G µ ( x ∗ + iy t n ,µ ( x ∗ )). Now, since y t n ,µ ( x ∗ ) → (cid:60) G µ ( x ∗ + iy t n ,µ ( x ∗ )) converges to the Hilbert transform (cid:90) dµ ( s ) x ∗ − s of µ at x ∗ , the integral being understood as a principal value integral. This proves x n − x ∗ t n = O ( t n ). Proof of Theorem 1.3 Throughout this section we assume Assumptions 1 and 2, t n is a sequence such that t n → nt κ − κn (log n ) ρ → ∞ , and nt κ − κn m n +1 → ∞ , as n → ∞ , and x ∗ is an interior point of thesupport of µ such that ψ ( x ) ∼ C | x − x ∗ | κ , as x → x ∗ , with 0 < κ < Auxiliary results For the large n asymptotics of A n , we need asymptotic equivalence of y t n ,µ ( x ∗ ) and y t n ,µ n ( x n ), where x n = (cid:60) z n . First, the asymptotic equivalence is shown for y t n ,µ ( x ∗ ) nd y t n ,µ n ( x ∗ ) (Lemma 4.5). By a Taylor expansion of y t n ,µ n ( x ∗ ) around x ∗ , this is trans-ferred to y t n ,µ n ( x n ) (Lemma 4.7). The necessary estimates on the derivative y (cid:48) t n ,µ n andthe difference | x n − x ∗ | are contained in Lemmas 4.4 and 4.6, respectively. Lemma 4.1. For every ε > there exist δ > and n ∈ N such that for every n > n and x ∈ [ x ∗ − δ, x ∗ + δ ] we have y t n ,µ n ( x ) ≥ (1 − ε ) (cid:18) Cκ + 1 (cid:19) − κ t − κ n . Proof. We define the sequence α n = (cid:16) (1 − ε ) Ct n κ +1 (cid:17) − κ for some ˜ ε ∈ (cid:0) , (cid:1) . Then we have nα κ +1 n → ∞ as n → ∞ . By (1.18) we have for any real x that (cid:90) x + α n x − α n dµ n ( s ) ≥ (cid:90) x + α n x − α n dµ ( s ) − m n n . Moreover, there exists a suitably small δ > x ∈ [ x ∗ − δ, x ∗ + δ ] and n > n (cid:90) x + α n x − α n dµ ( s ) ≥ (cid:18) − ˜ ε (cid:19) (cid:90) x ∗ + α n x ∗ − α n dµ ( s ) ≥ − ˜ ε ) Cα κ +1 n κ + 1 . This gives1 n n (cid:88) k =1 α n + ( a ( n ) k − x ) ≥ α n (cid:90) x + α n x − α n dµ n ( s ) ≥ α n (cid:26) − ˜ ε ) Cα κ +1 n κ + 1 − m n n (cid:27) ≥ (1 − ε ) Cκ + 1 α κ +1 n = 1 t n . Here we used the fact that nt κ − κn m n +1 → ∞ and hence ˜ m n n = o ( α κ +1 n ) as n → ∞ . It followsthat y t n ,µ n ( x ) ≥ α n for n > n and x ∈ [ x ∗ − δ, x ∗ + δ ]. The statement now easily followsby choosing a suitably small ˜ ε . (cid:3) Lemma 4.2. Let α ∈ R and β > . Then there exist constants K, K (cid:48) > such that |(cid:61) G µ ( x ∗ + α + iβ ) | ≤ K | α | κ + K (cid:48) β κ , as α + iβ → .Proof. Let µ be supported on [ a, b ]. Then we obtain for some constant K > |(cid:61) G µ ( x ∗ + α + iβ ) | ≤ K β (cid:90) ba | s − x ∗ | κ ( x ∗ − s + α ) + β ds. Assuming without loss of generality α > |(cid:61) G µ ( x ∗ + α + iβ ) | ≤ K β (cid:40) (cid:90) x ∗ a | s − x ∗ | κ ( x ∗ − s ) + β ds + (cid:90) x ∗ + αx ∗ | s − x ∗ | κ ( x ∗ + α − s ) + β ds + (cid:90) bx ∗ + α | s − x ∗ | κ ( x ∗ + α − s ) + β ds (cid:41) . For the first integral we have for some constant K > (cid:90) x ∗ a | s − x ∗ | κ ( x ∗ − s ) + β ds = β κ − (cid:90) a − x ∗ β | s | κ s + 1 ≤ K β κ − . or the second integral we have for some constant K > (cid:90) x ∗ + αx ∗ | s − x ∗ | κ ( x ∗ + α − s ) + β ds = α κ − (cid:90) s κ (1 − s ) + (cid:0) βα (cid:1) ds ≤ K α κ β . Finally, for the third integral we obtain for some constant K > (cid:90) bx ∗ + α | s − x ∗ | κ ( x ∗ + α − s ) + β ds ≤ (cid:90) bx ∗ + α | s − x ∗ − α | κ + α κ ( x ∗ + α − s ) + β ds ≤ K (cid:18) α κ β + β κ − (cid:19) , which proves the statement. (cid:3) Lemma 4.3. For every constant K > we have lim sup n →∞ sup x ∈ (cid:20) x ∗ − Kt − κn , x ∗ + Kt − κn (cid:21) y t n ,µ n ( x ) t − κ n < ∞ . In other words, we have y t n ,µ n ( x ) = O ( t − κ n ) , if x − x ∗ = O ( t − κ n ) , as n → ∞ .Proof. By Lemma 3.1 and Lemma 4.1 we have y t n ,µ n ( x ) πt n = − π (cid:61) G µ n ( x + iy t n ,µ n ( x )) = − π (cid:61) G µ ( x + iy t n ,µ n ( x )) + O (cid:32) m n + 1 nt − κ n (cid:33) , as n → ∞ . Now, using Lemma 4.2 we obtain y t n ,µ n ( x ) = O (cid:32) t n [( x − x ∗ ) κ + y t n ,µ n ( x ) κ ] + m n + 1 nt κ − κ n (cid:33) = O (cid:16) t n t κ − κ n + t n y t n ,µ n ( x ) κ (cid:17) , as n → ∞ . Dividing this by y t n ,µ n ( x ) κ gives y t n ,µ n ( x ) − κ = O ( t n ) , as n → ∞ , from which the statement follows. (cid:3) Lemma 4.4. For every constant K > we have lim sup n →∞ sup x ∈ (cid:20) x ∗ − Kt − κn , x ∗ + Kt − κn (cid:21) | y (cid:48) t n ,µ n ( x ) | < ∞ . In other words, we have y (cid:48) t n ,µ n ( x ) = O (1) , if x − x ∗ = O ( t − κ n ) , as n → ∞ . roof. If x − x ∗ = O ( t − κ n ), as n → ∞ , we have for positive constants C j ( j = 1 , . . . , n large enough | H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) | ≥ (cid:60) H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) = 2 t n y t n ,µ n ( x ) (cid:90) dµ n ( s ) | x − s + iy t n ,µ n ( x ) | ≥ t n y t n ,µ n ( x ) (cid:90) x + y tn,µn ( x ) x − y tn,µn ( x ) dµ n ( s )(( x − s ) + y t n ,µ n ( x ) ) ≥ t n y t n ,µ n ( x ) (cid:90) x + y tn,µn ( x ) x − y tn,µn ( x ) dµ n ( s ) ≥ t n y t n ,µ n ( x ) (cid:40)(cid:90) x + y tn,µn ( x ) x − y tn,µn ( x ) dµ ( s ) − C ˜ m n n (cid:41) ≥ t n y t n ,µ n ( x ) (cid:40) (cid:90) x ∗ + y tn,µn ( x ) x ∗ − y tn,µn ( x ) dµ ( s ) − C ˜ m n n (cid:41) ≥ t n y t n ,µ n ( x ) (cid:26) C y t n ,µ n ( x ) κ +1 − C ˜ m n n (cid:27) = C t n y t n ,µ n ( x ) κ − − C t n ˜ m n ny t n ,µ n ( x ) ≥ C > , where in the last estimates we used the Lemmas 4.1 and 4.3. Moreover, we have | H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) | ≤ | y (cid:48) t n ,µ n ( x ) | = | tan (cid:0) − arg H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) (cid:1) | . From the above computation we know that arg H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) ∈ (cid:2) − π + ε, π − ε (cid:3) for some small ε > 0, so that the statement follows. (cid:3) Lemma 4.5. We have y t n ,µ ( x ∗ ) ∼ (cid:32) Cπt n sin (cid:0) π κ (cid:1) (cid:33) − κ , n → ∞ , and y t n ,µ n ( x ∗ ) ∼ (cid:32) Cπt n sin (cid:0) π κ (cid:1) (cid:33) − κ , n → ∞ . Proof. We have for a sequence δ n → t n δ n → n → ∞ t n (cid:90) dµ ( s )( s − x ∗ ) + y t n ,µ ( x ∗ ) = t n (cid:90) x ∗ + δ n x ∗ − δ n dµ ( s )( s − x ∗ ) + y t n ,µ ( x ∗ ) + o (1) , as n → ∞ . Moreover, we have t n (cid:90) x ∗ + δ n x ∗ − δ n dµ ( s )( s − x ∗ ) + y t n ,µ ( x ∗ ) ∼ Ct n (cid:90) δ n − δ n | s | κ s + y t n ,µ ( x ∗ ) ds = Ct n y t n ,µ ( x ∗ ) κ − (cid:90) δ n /y tn,µ ( x ∗ ) − δ n /y tn,µ ( x ∗ ) | s | κ s + 1 ds. Since δ n y tn,µ ( x ∗ ) → ∞ , as n → ∞ , and using the identity (cid:90) ∞−∞ | s | κ s + 1 ds = π sin (cid:0) π κ +12 (cid:1) the first part of the statement follows. For the second part we consider = t n (cid:90) dµ n ( s )( s − x ∗ ) + y t n ,µ n ( x ∗ ) = − t n y t n ,µ n ( x ∗ ) (cid:61) G µ n (cid:0) x ∗ + iy t n ,µ n ( x ∗ ) (cid:1) = t n (cid:90) dµ ( s )( s − x ∗ ) + y t n ,µ n ( x ∗ ) − t n y t n ,µ n ( x ∗ ) {(cid:61) G µ n ( x ∗ + iy t n ,µ n ( x ∗ )) − (cid:61) G µ ( x ∗ + iy t n ,µ n ( x ∗ )) } . By Lemma 3.1 and Lemma 4.1 we have (cid:12)(cid:12)(cid:12)(cid:12) t n y t n ,µ n ( x ∗ ) {(cid:61) G µ n ( x ∗ + iy t n ,µ n ( x ∗ )) − (cid:61) G µ ( x ∗ + iy t n ,µ n ( x ∗ )) } (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:32) ˜ m n nt κ − κ n (cid:33) = o (1) , as n → ∞ . Hence, for a sequence δ n → t n δ n → 0, as n → ∞ , we have1 = t n (cid:90) x ∗ + δ n x ∗ − δ n dµ ( s )( x ∗ − s ) + y t n ,µ n ( x ∗ ) + o (1) , as n → ∞ . From here we can proceed as in the first part of the proof to obtain thesecond statement. (cid:3) For the next Lemmas we recall that by definition of the saddle points we have for real u, v : z n = F t n ,µ n (cid:18) H t n ,µ ( x ∗ + y t n ,µ ( x ∗ )) + unc t n (cid:19) ,w n = F t n ,µ n (cid:18) H t n ,µ ( x ∗ + y t n ,µ ( x ∗ )) + vnc t n (cid:19) ,x n = (cid:60) z n , (cid:61) z n = y t n ,µ n ( x n ) , with c t n = ψ t n ( x ∗ t n ) = y t n ,µ ( x ∗ ) πt n . Lemma 4.6. If u and v belong to a compact subset, then there exist positive constants K, K (cid:48) and n ∈ N independent of u and v , such that for n > n we have | x n − x ∗ | ≤ K ˜ m n nt κ − κ n and | z n − w n | ≤ K (cid:48) ˜ m n nt κ − κ n . Thus, we have x n − x ∗ = O (cid:18) m n +1 nt κ − κn (cid:19) and z n − w n = O (cid:18) m n +1 nt κ − κn (cid:19) , as n → ∞ , uniformlywith respect to u, v on compact subsets.Proof. We split the proof in four parts.(i) We first show | y t n ,µ ( x ∗ ) − y t n ,µ n ( x ∗ ) | = O (cid:18) ˜ m n t n n (cid:19) , n → ∞ . o see that, we observe that we have as n → ∞ (see the proof of Lemma 4.5)1 t n = (cid:90) dµ ( s )( x ∗ − s ) + y t n ,µ ( x ∗ ) = (cid:90) dµ ( s )( x ∗ − s ) + y t n ,µ n ( x ∗ ) + O (cid:32) ˜ m n nt − κ n (cid:33) , n → ∞ . This gives (cid:90) x ∗ − s ) + y t n ,µ ( x ∗ ) − x ∗ − s ) + y t n ,µ n ( x ∗ ) dµ ( s )= | y t n ,µ ( x ∗ ) − y t n ,µ n ( x ∗ ) | (cid:90) dµ ( s )(( x ∗ − s ) + y t n ,µ ( x ∗ ) ) (( x ∗ − s ) + y t n ,µ n ( x ∗ ) )= O (cid:32) ˜ m n nt − κ n (cid:33) , n → ∞ . Using Lemma 4.5, by an elementary estimation we have for n large enough and apositive constant c (cid:90) dµ ( s )(( x ∗ − s ) + y t n ,µ ( x ∗ ) ) (( x ∗ − s ) + y t n ,µ n ( x ∗ ) ) ≥ c t n t − κ n , which gives | y t n ,µ ( x ∗ ) − y t n ,µ n ( x ∗ ) | = O (cid:18) ˜ m n t n n (cid:19) , n → ∞ . (ii) Next we show H t n ,µ ( x ∗ + iy t n ,µ ( x ∗ )) + unc t n − H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )) = O (cid:32) ˜ m n nt κ − κ n (cid:33) , n → ∞ . To this end, we write H t n ,µ ( x ∗ + iy t n ,µ ( x ∗ )) + unc t n − H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ ))= unc t n + t n (cid:60) { G µ ( x ∗ + y t n ,µ ( x ∗ )) − G µ n ( x ∗ + y t n ,µ n ( x ∗ )) } = unc t n + t n (cid:60) { G µ ( x ∗ + y t n ,µ n ( x ∗ )) − G µ n ( x ∗ + y t n ,µ n ( x ∗ )) } + t n (cid:60) { G µ ( x ∗ + y t n ,µ ( x ∗ )) − G µ ( x ∗ + y t n ,µ n ( x ∗ )) } . Now, by Lemmas 3.1 and 4.5 we have t n (cid:60) { G µ ( x ∗ + y t n ,µ n ( x ∗ )) − G µ n ( x ∗ + y t n ,µ n ( x ∗ )) } = O (cid:32) ˜ m n nt κ − κ n (cid:33) , n → ∞ . Moreover, we obtain t n (cid:60) { G µ ( x ∗ + y t n ,µ ( x ∗ )) − G µ ( x ∗ + y t n ,µ n ( x ∗ )) } = t n (cid:0) y t n ,µ ( x ∗ ) − y t n ,µ n ( x ∗ ) (cid:1) (cid:90) ( x ∗ − s ) dµ ( s )(( x ∗ − s ) + y t n ,µ ( x ∗ ) ) (( x ∗ − s ) + y t n ,µ n ( x ∗ ) ) , and the last integral can be seen using straightforward estimates to be O (cid:18) t n t − κn (cid:19) ,so that it tends to 0 as n → ∞ . Thus, using (i) we obtain (ii). iii) By the proof of Lemma 4.4 we have for some constant c > (cid:60) H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) ≥ c , |(cid:61) H (cid:48) t n ,µ n ( x + iy t n ,µ n ( x )) | ≤ , for x − x ∗ = O ( t − κ n ) , as n → ∞ . This gives for some constant c > (cid:60) (cid:26) ddx H t n ,µ n ( x + iy t n ,µ n ( x )) (cid:27) ≥ c for n large enough, if x − x ∗ = O ( t − κ n ) , as n → ∞ .(iv) Now, we choose the constant K > u in a given compact set wehave | H t n ,µ ( x ∗ + iy t n ,µ ( x ∗ )) + unc t n − H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )) | ≤ K ˜ m n nt κ − κ n . If we define δ n = K ˜ m n +1 c nt κ − κn , then using (iii) we know that by x (cid:55)→ H t n ,µ n ( x + iy t n ,µ n ( x )) the interval ( x ∗ − δ n , x ∗ + δ n ) is mapped bijectively onto an intervalcontaining (cid:34) H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )) − K ˜ m n nt κ − κ n , H t n ,µ n ( x ∗ + iy t n ,µ n ( x ∗ )) + K ˜ m n nt κ − κ n (cid:35) . As this interval contains the point H t n ,µ ( x ∗ + iy t n ,µ ( x ∗ )) + unc tn , we can concludethat for its preimage we have | x n − x ∗ | ≤ δ n = O (cid:18) ˜ m n nc t n (cid:19) , as n → ∞ uniformly with respect to u . From this and Lemma 4.4 we additionallyobtain | y t n ,µ n ( x n ) − y t n ,µ n ( x ∗ ) | = O (cid:18) ˜ m n nc t n (cid:19) , n → ∞ , which gives us z n − ( x ∗ + y t n µ n ( x ∗ )) = O (cid:18) ˜ m n nc t n (cid:19) , n → ∞ . From this it easily follows that z n − w n = O (cid:18) ˜ m n nc t n (cid:19) = O (cid:32) ˜ m n nt κ − κ n (cid:33) , n → ∞ , uniformly with respect to u and v in compact subsets. (cid:3) Lemma 4.7. We have uniformly in u on compact subsets y t n ,µ n ( x n ) ∼ (cid:32) Cπt n sin (cid:0) π κ (cid:1) (cid:33) − κ , n → ∞ . Proof. From (iv) in the proof of Lemma 4.6 we know that | y t n ,µ n ( x n ) − y t n ,µ n ( x ∗ ) | = O (cid:18) ˜ m n t n n (cid:19) , n → ∞ . sing Lemma 4.5 and recalling that we have c t n = y tn,µ ( x ∗ ) πt n , we obtain | y t n ,µ n ( x n ) − y t n ,µ n ( x ∗ ) | = O (cid:32) ˜ m n t n nt − κ n (cid:33) , n → ∞ . Dividing this by y t n ,µ n ( x ∗ ) und using Lemma 4.5 we obtain y t n ,µ n ( x n ) y t n ,µ n ( x ∗ ) = 1 + o (1) , as n → ∞ , uniformly in u on compact subsets, from which the statement follows. (cid:3) Lemma 4.8. We have uniformly with respect to u and v on compact subsets lim n →∞ φ n ( z n , u ) − φ n ( w n , v ) − v − ut n c t n (cid:0) z n − x ∗ t n (cid:1) = 0 . Proof. We first recall that by definition the phase function is given by φ n ( z, u ) = n t n (cid:34)(cid:18) z − x ∗ t n − unc t n (cid:19) + 2 t n (cid:90) log( z − s ) dµ n ( s ) (cid:35) . Hence, we obtain φ n ( z n , u ) − φ n ( w n , v ) = φ n ( z n , u ) − φ n ( z n , v ) + φ n ( z n , v ) − φ n ( w n , v )= n t n (cid:34)(cid:18) z n − x ∗ t n − unc t n (cid:19) − (cid:18) z n − x ∗ t n − vnc t n (cid:19) (cid:35) + (cid:90) z n w n φ (cid:48) n ( s, v ) ds = v − ut n c t n (cid:0) z n − x ∗ t n (cid:1) − ( v − u )( u + v )2 t n c t n n + nt n (cid:90) z n w n ( H t n ,µ n ( s ) − H t n ,µ n ( w n )) ds, where the integral is performed over the part of the graph of y t n ,µ n from w n to z n . Nowwe have ( v − u )( u + v )2 t n c t n n → , n → ∞ , uniformly in u and v , and (cid:12)(cid:12)(cid:12)(cid:12) nt n (cid:90) z n w n ( H t n ,µ n ( s ) − H t n ,µ n ( w n )) ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) nt n (cid:90) z n w n (cid:90) sw n H (cid:48) t n ,µ n ( x ) dxds (cid:12)(cid:12)(cid:12)(cid:12) , which by Lemma 4.4 can be bounded above, so that we obtain (cid:12)(cid:12)(cid:12)(cid:12) nt n (cid:90) z n w n ( H t n ,µ n ( s ) − H t n ,µ n ( w n )) ds (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) nt n ( z n − w n ) max | H (cid:48) t n ,µ n ( z ) | (cid:19) , as n → ∞ , where the maximum is taken over the part of the graph of y t n ,µ n from w n to z n . By the boundedness of H (cid:48) t n ,µ n ( z ) and using Lemma 4.7 this can be bounded further,and obtain (cid:12)(cid:12)(cid:12)(cid:12) nt n (cid:90) z n w n ( H t n ,µ n ( s ) − H t n ,µ n ( w n )) ds (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:18) t n c t n n (cid:19) = o (1) , n → ∞ , uniformly in u and v on compact sets, from which the statement follows. (cid:3) .2. Asymptotics of A n In order to prove that lim n →∞ A n ( u, v ) = sin ( π ( u − v )) π ( u − v )uniformly with respect to u and v on compact subsets, we have to provelim n →∞ st n c t n = π, with s = y t n ,µ n ( x n ). However, recalling that c t n = y tn,µ ( x ∗ ) πt n , this immediately follows by acombination of Lemma 4.5 and Lemma 4.7. Estimation of I n Using the notations and assumptions in the statement of Theorem 1.3 we recall that theintegral I n is given by I n ( u, v ) = 1 t n c t n e − u − v ntnc tn + u − vtnctn ( x n − x ∗ tn ) 1(2 πi ) (cid:90) x n + i ∞ x n − i ∞ dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w . As done previously, we split up the integral into seven parts, of which we explicitlyhave to deal with ˜ I (1) n = (cid:90) z n + i ∞ z n + iL n dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w , ˜ I (4) n = (cid:90) z n + iL n z n − L n dz (cid:90) γ n ∩ U Ln ( w n ) dw e φ n ( z,u ) − φ n ( w,v ) z − w , ˜ I (5) n = (cid:90) z n + iL n z n − L n dz (cid:90) γ n \ U Ln ( w n ) dw e φ n ( z,u ) − φ n ( w,v ) z − w , where L n is defined by L n = (cid:114) t n n (log n ) ρ for some ρ > 0. Using the assumption nt κ − κn (log n ) ρ → ∞ , as n → ∞ , we havelim n →∞ L n t n c t n = lim n →∞ L n t − κ n = 0 . Moreover, we observe (see the proof of Lemma 4.4) that the sequence t n n β n = t n n (cid:60) φ (cid:48)(cid:48) n ( z n , u ) = 2 t n y t n ,µ n ( x n ) (cid:90) dµ n ( s ) | z n − s | is bounded in absolute value and stays away from zero as n → ∞ . Following the strategyof the proofs of Theorem 1.1 and Theorem 1.2 we will have to showlim n →∞ t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I ( j ) n = 0 , j = 1 , , , locally uniformly in u and v . stimation of ˜ I (1) n . First we observe that by Lemma 4.4 we have for some constant K > d − n = 1min | z − w | ≤ K L n , where the minimum is taken on the contour [ z n + iL n , z n + i ∞ ] × γ n . This gives (cid:12)(cid:12)(cid:12)(cid:12) t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (1) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ K t n c t n L n L ( γ n ) e u − vtnctn ( x n − x ∗ tn ) (cid:90) z n + i ∞ z n + iL n e (cid:60) φ n ( z,u ) | dz | (cid:90) γ n e −(cid:60) φ n ( w,v ) | dw | . The length of γ n can be estimated by Lemma 2.1 and we obtain further, using Lemma4.8, for some constant K > (cid:12)(cid:12)(cid:12)(cid:12) t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (1) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ K t n c t n L n √ t n n e u − vtnctn ( x n − x ∗ tn ) + (cid:60){ φ n ( z n ,u ) − φ n ( w n ,v ) } (cid:90) z n + i ∞ z n + iL n e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } | dz |≤ K t n c t n L n √ t n n e (cid:60){ φ n ( z n + iL n ,u ) − φ n ( z n ,u ) } (cid:90) z n + i ∞ z n + iL n e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } | dz | . It follows from elementary considerations that the integral (cid:90) z n + i ∞ z n + iL n e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } | dz | remains bounded as n → ∞ , so that it is sufficient to show (cid:60) { φ n ( z n + iL n , u ) − φ n ( z n , u ) } ≤ − K (log n ) ρ for some constant K > n large enough. A complex Taylor expansion for φ n ( z, u )around z n yields φ n ( z, u ) = φ n ( z n , u ) + 12 φ (cid:48)(cid:48) n ( z n , u )( z − z n ) + R ( z ) . We will estimate R ( z ) for | z − z n | ≤ L n in a similar way than in the estimation of ˜ I (1) n in the proof of Theorem 1.2. To this end, we proceed as in (3.2) to obtain | R ( z ) | ≤ max | φ (cid:48)(cid:48)(cid:48) n ( ξ ; u ) | L n where the maximum is over the line segment between z n and z . Given ξ on this linesegment, we let γ be the circle of radius (cid:61) ξ ≥ ( (cid:61) z n − L n ) around ξ , and we again useCauchy’s theorem to estimate the maximum, | φ (cid:48)(cid:48)(cid:48) n ( ξ ; u ) | = (cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) γ φ (cid:48)(cid:48) n ( ζ ; u )( ζ − ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:61) ξ max ζ ∈ γ | φ (cid:48)(cid:48) n ( ζ ; u ) | ≤ nt n ( (cid:61) z n − L n ) max ζ ∈ γ | t n G (cid:48) µ n ( ζ ) | . Now, for any ξ on the line segment between z n and z , we have for ζ ∈ γ that |(cid:60) ζ − (cid:60) z n | ≤ L n + 32 ( (cid:61) z n + L n ) , so that by Lemma 4.6 we have |(cid:60) ζ − x ∗ | ≤ K t − κ n , here the constant K > ζ and ξ (and u ). Hence, by Lemma4.3 there is a constant K > n we have y t n ,µ n ( (cid:60) ζ ) ≤ K t − κ n uniformly in ζ, ξ and u . But then we find a small constant ˆ c > n uniformly y ˆ ct n ,µ n ( (cid:60) ζ ) ≤ (cid:61) ζ. This gives | G (cid:48) µ n ( ζ ) | ≤ ct n , for large n uniformly. This yields for ξ ∈ U L n ( z n ) | φ (cid:48)(cid:48)(cid:48) n ( ξ ; u ) | ≤ K nt − κ n for a constant K > 0, so that for | z − z n | ≤ L n we obtain | R ( z ) | ≤ K nt − κ n L n = o (cid:0) (log n ) ρ (cid:1) , as n → ∞ . From this we obtain (cid:60) { φ n ( z n + iL n , u ) − φ n ( z n , u ) } ≤ − K (log n ) ρ for some constant K > n large enough, which is sufficient to showlim n →∞ t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (1) n = 0 , uniformly in u and v on compact subsets. Estimation of ˜ I (4) n . From Lemma 4.4 we know that the length of γ n ∩ U L n ( w n ) is O ( L n )as n → ∞ , and that the angle between γ n and the vertical line z n + i R cannot becomesmall for large n . Hence, we can proceed as in the estimation of ˜ I (4) n in the proofs ofTheorems 1.1 and 1.2, which means that we obtain (cid:12)(cid:12)(cid:12)(cid:12) t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (4) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ L n t n c t n e u − vtnctn ( x n − x ∗ tn ) + (cid:60){ φ n ( z n ,u ) − φ n ( w n ,v ) } . Using Lemma 4.8 the expression on the right-hand side vanishes as n → ∞ locallyuniformly in u and v . Estimation of ˜ I (5) n . By Lemma 4.6 we have z n − w n = o ( L n ) as n → ∞ , locally uniformlyin u and v , which means that we again obtain˜ d n − = 1min | z − w | ≤ K L n for some constant K > 0, where the minimum is taken over the contour [ z n − iL n , z n + iL n ] × ( γ n \ U L n ( w n )). Hence, we obtain (cid:12)(cid:12)(cid:12)(cid:12) t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (5) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ K t n c t n L n e u − vtnctn ( x n − x ∗ tn ) + (cid:60){ φ n ( z n ,u ) − φ n ( w n ,v ) } × (cid:90) z n + iL n z n − iL n e (cid:60){ φ n ( z,u ) − φ n ( z n ,u ) } | dz | (cid:90) γ n \ U Ln ( w n ) e (cid:60){ φ n ( w n ,v ) − φ n ( w,v ) } | dz | , hich, by Lemma 4.8 can be estimated further to (cid:12)(cid:12)(cid:12)(cid:12) t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (5) n (cid:12)(cid:12)(cid:12)(cid:12) ≤ K t n c t n (cid:90) γ n \ U Ln ( w n ) e (cid:60){ φ n ( w n ,v ) − φ n ( w,v ) } | dz | for some constant K > 0. Now defining { w + n , w − n } = γ n ∩ ∂U L n ( w n ) the last integral canbe estimated above by L ( γ n ) (cid:110) e (cid:60) { φ n ( w n ,v ) − φ n ( w + n ,v ) } + e (cid:60) { φ n ( w n ,v ) − φ n ( w − n ,v ) } (cid:111) . The length of γ n can be estimated by Lemma 2.1, whereas the exponential terms can bebounded the same way as in the estimation of ˜ I (1) n above. This finally leads to (cid:12)(cid:12)(cid:12)(cid:12) t n c t n e u − vtnctn ( x n − x ∗ tn ) ˜ I (5) n (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) n / e − K (log n ) ρ (cid:17) , as n → ∞ , uniformly in u and v on compact subsets, for some constant K > 0. Aspreviously seen in the proofs of Theorems 1.1 and 1.2, this completes the proof of Theorem1.3. Proof of Theorem 1.4 We start with some elementary lemmas. Lemma 5.1. Let µ be a probability measure on the real line, x ∗ ∈ R , t > , and let x ∗ t be defined by (2.3) , Then, | x ∗ t − x ∗ | ≤ √ t. Proof. If z ∈ Ω t,µ , it follows from (1.11) and the Cauchy-Schwarz inequality that theStieltjes transform G µ ( z ) satisfies the bound | G µ ( z ) | ≤ (cid:90) dµ ( s ) | z − s | ≤ (cid:18)(cid:90) dµ ( s ) | z − s | (cid:19) / ≤ √ t . It follows from (2.1) that sup z ∈ Ω t,µ | H t,µ ( z ) − z | ≤ √ t. (5.1)Applying this to z = x ∗ + iy t,µ ( x ∗ ) and using (2.3), we get | H t,µ ( x ∗ + iy t,µ ( x ∗ )) − x ∗ − iy t,µ ( x ∗ ) | ≤ √ t, and the result now follows easily. (cid:3) Lemma 5.2. Let µ, ν be probability measures on the real line, let x ∗ ∈ R , and let x ∗ t bedefined by (2.3) . We have for any ε ∈ R , t > that |(cid:60) { F t,ν ( x ∗ t + ε ) } − x ∗ | ≤ √ t + | ε | . Proof. It follows from (5.1) applied to ν and z = F t,ν ( x ∗ t + ε ) that | F t,ν ( x ∗ t + ε ) − x ∗ t − ε | ≤ √ t. By Lemma 5.1, it follows that | F t,ν ( x ∗ t + ε ) − x ∗ | ≤ √ t + | ε | . (cid:3) emma 5.3. Let x ∗ ∈ R , let δ n , ε n , t n be sequences of positive numbers converging to ,as n → ∞ , and let µ n be a sequence of probability measures. Define, for u, v ∈ R , z n = F t n ,µ n (cid:0) x ∗ t n + ε n u (cid:1) , w n = F t n ,µ n (cid:0) x ∗ t n + ε n v (cid:1) . (5.2) If [ x ∗ − δ n , x ∗ + δ n ] does not intersect with the support of µ n , if ε n = o ( δ n ) , and if t n = o ( δ n ) as n → ∞ , we have (cid:61) z n = (cid:61) w n = 0 , | z n − w n | ≤ ε n | u − v | (5.3) for n sufficiently large, locally uniformly in u, v .Proof. Because of the conditions imposed on the sequences δ n , ε n , t n , we have by Lemma5.2 that |(cid:60) z n − x ∗ | = o ( δ n ) as n → ∞ , and it follows that [ (cid:60) z n − δ n / , (cid:60) z n + δ n / 2] doesnot intersect with the support of µ n for n sufficiently large. Similar estimates hold for w n . Furthermore, if x ∈ R is such that dist( x, supp( µ n )) ≥ √ t n , we have y t n ,µ n ( x ) = 0 , and this implies that z n , w n ∈ R . Next, it is easy to see that (cid:12)(cid:12) G (cid:48) µ n ( z n ) (cid:12)(cid:12) ≤ δ n , (cid:12)(cid:12) G (cid:48) µ n ( w n ) (cid:12)(cid:12) ≤ δ n (5.4)for n sufficiently large. We also have | G µ n ( z n ) − G µ n ( w n ) | = | z n − w n | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dµ n ( s )( z n − s )( w n − s ) (cid:12)(cid:12)(cid:12)(cid:12) . The latter can be estimated using the Cauchy-Schwarz inequality by ≤ | z n − w n | (cid:115)(cid:90) dµ n ( s )( z n − s ) (cid:90) dµ n ( s )( w n − s ) = | z n − w n | (cid:113) G (cid:48) µ n ( z n ) G (cid:48) µ n ( w n ) , giving | G µ n ( z n ) − G µ n ( w n ) | ≤ | z n − w n | δ − n , for n sufficiently large. By the definition of z n and w n we have z n − x ∗ t n − ε n u + t n G µ n ( z n ) = 0 ,w n − x ∗ t n − ε n v + t n G µ n ( w n ) = 0 , so that | z n − w n | ≤ ε n | u − v | + t n | G µ n ( z n ) − G µ n ( w n ) | ≤ ε n | u − v | + 4 t n δ − n | z n − w n | , which implies (5.3) for n large, since t n δ − n → n → ∞ . (cid:3) For the correlation kernel K n,t , recall (1.9), which gives ε n K n,t n (cid:0) x ∗ t n + ε n u, x ∗ t n + ε n v (cid:1) = A n ( u, v ) + I n ( u, v ) , with I n ( u, v ) = e − nε n tn ( u − v )+ nεntn ( u − v )( (cid:60) z n − x ∗ ) nε n (2 πi ) t n (cid:60) z n + i ∞ (cid:90) (cid:60) z n − i ∞ dz (cid:90) γ n dw e φ n ( z,u ) − φ n ( w,v ) z − w , where φ n ( z, u ) = n t n (cid:8) ( z − x ∗ t n − ε n u ) + 2 t n g µ n ( z ) (cid:9) , n consists of the graph y t n ,µ n and its complex conjugate (positively oriented), and A n ( u, v ) = 1 π ( u − v ) sin (cid:18) nε n st n ( u − v ) (cid:19) , where s = (cid:61) (cid:8) F t n ,µ n (cid:0) x ∗ t n + ε n u (cid:1)(cid:9) = (cid:61) z n and x = (cid:60) z n . Under the conditions of Lemma5.3, z n is real for large n so that A n ( u, v ) = 0 . Similarly as in the proof of Lemma 4.8, we have φ n ( z, u ) − φ n ( z, v ) = nε n t n ( u − v ) − nε n t n ( u − v )( z − x ∗ ) , and it follows that we can write I n ( u, v ) as I n ( u, v ) = nε n (2 πi ) t n z n + i ∞ (cid:90) z n − i ∞ dz (cid:90) γ n dw e φ n ( z,v ) − φ n ( w,v ) − nεntn ( u − v )( z − z n ) z − w . Thus, in order to see thatlim n →∞ ε n K n,t n (cid:0) x ∗ t n + ε n u, x ∗ t n + ε n v (cid:1) = 0 , it is sufficient to show lim n →∞ I n ( u, v ) = 0 . The critical points of φ n ( z, v ) and φ n ( w, v ) are defined by the equations H t n ,µ n ( z ) = x ∗ t n + ε n v, H t n ,µ n ( w ) = x ∗ t n + ε n v, which means that they are both precisely the real point w n defined in (5.2). If ε n = o ( δ n ) and t n = o ( δ n ) as n → ∞ , it follows from Lemma 5.2 that w n lies in an interval[ x ∗ − δ n / , x ∗ + δ n / 4] for n sufficiently large.It is crucial to observe that the parts of γ n lying on the real line do not contribute to theintegral I n ( u, v ), as they cancel out against their complex conjugate. As a consequenceof this observation and by symmetry with respect to complex conjugation, in order toshow that the large n limit of I n ( u, v ) is zero, it is sufficient to show thatlim n →∞ nε n t n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z n + i ∞ (cid:90) z n dz (cid:90) γ ± n dw e φ n ( z,v ) − φ n ( w,v ) − nεntn ( u − v )( z − z n ) z − w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (5.5)where γ ± n denotes the part of γ n which lies to the right (+) or left ( − ) of w n and whichis not located on the real axis. For w ∈ γ ± n we have |(cid:60) w − (cid:60) w n | ≥ δ n / n sufficientlylarge by construction.Using the inequality | z − w | ≥ δ n / z, w ) ∈ ( z n + i R ) × γ ± n , we have nε n t n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z n + i ∞ (cid:90) z n dz (cid:90) γ ± n dw e φ n ( z,v ) − φ n ( w,v ) − nεntn ( u − v )( z − z n ) z − w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nε n t n δ n I (1) n × I (2) n , (5.6) ith I (1) n = z n + i ∞ (cid:90) z n e (cid:60) ( φ n ( z,v ) − φ n ( z n ,v )) | dz | ,I (2) n = (cid:90) γ ± n e (cid:60) ( φ n ( w n ,v ) − φ n ( w,v )) | dw | . We set L n = √ t n log n and we will now estimate these integrals. Lemma 5.4. Let us assume the conditions of Lemma 5.3, and additionally let nδ n → + ∞ , as n → ∞ . Then, locally uniformly in v , we have I (2) n ≤ | γ ± n | e − n tn L n (5.7) for n sufficiently large, where | γ ± n | denotes the length of the curve γ ± n .Proof. We give the proof for γ + n , the case of γ − n is similar. We easily obtain the bound I (2) n ≤ e max w ∈ γ + n (cid:60) ( φ n ( w n ,v ) − φ n ( w,v )) | γ + n | . Next, we note that for x ∈ [ w n − L n , w n + L n ], dist( x, supp( µ n )) ≥ √ t n , for n large,which means that y t n ,µ n ( x ) = 0 . Hence, [ w n − L n , w n + L n ] is part of γ n , but not of γ + n . Since γ n is a path of descentfor − φ n ( w, v ), we obtain that −(cid:60) φ n ( w, v ) attains its maximum on γ + n at a real point w ≥ w n + L n , and we have e (cid:60) ( φ n ( w n ,v ) − φ n ( w,v )) ≤ e (cid:60) ( φ n ( w n ,v ) − φ n ( w n + L n ,v )) for w ∈ γ + n .It follows that I (3) n ≤ | γ + n | e (cid:60) ( φ n ( w n ,v ) − φ n ( w n + L n ,v )) . (5.8)Moreover, we have φ (cid:48)(cid:48) n ( z, v ) = nt n + nG (cid:48) µ n ( z ) . By (5.4), we have φ (cid:48)(cid:48) n ( z n , v ) ∼ nt n , φ (cid:48)(cid:48) n ( w n , v ) ∼ nt n , as n → ∞ . For | z − x ∗ | ≤ δ n / | g µ n ( z ) | ≤ log 2 δ n , for n sufficiently large, with the branches of the function g µ n ( z ) = (cid:90) log( z − s ) dµ n ( s )defined such that g µ n is analytic in the disk centered at x ∗ with radius δ n . If | w − w n | ≤ L n ,we have φ n ( w, v ) = φ n ( w n , v ) + 12 φ (cid:48)(cid:48) n ( w n , v )( w − w n ) + R ( w ) , here | R ( w ) | ≤ 64 max | w n − s | = δ n / | φ n ( s, v ) | δ n | w − w n | − | w − w n | δ n ≤ 64 max | x ∗ − s | = δ n / | φ n ( s, v ) || w − w n | δ n (1 − | w − w n | δ n ) , which, by use of Lemma 5.1, can be estimated for large n by O (cid:32) n log δ n + nδ n t n δ n L n (cid:33) = o (cid:18) nL n t n (cid:19) , n → ∞ , where in the last equality we used the assumption nδ n → ∞ , as n → ∞ . Using this in(5.8), we obtain (5.7). (cid:3) Lemma 5.5. Let us assume the conditions of Lemma 5.3. Then, locally uniformly in u, v , we have I (1) n = O (cid:0) √ t n (cid:1) , as n → ∞ .Proof. We have I (1) n = z n + i ∞ (cid:90) z n e −(cid:60) ( φ n ( z n ,v ) − φ n ( z,v )) | dz | = z n + i ∞ (cid:90) z n + e − n tn (cid:60) (cid:110) ( z n − x ∗ tn − ε n v ) − ( z − x ∗ tn − ε n v ) +2 t n g µn ( z n ) − t n g µn ( z ) (cid:111) | dz |≤ ∞ (cid:90) e − n tn (cid:60) (cid:110) ( z n − x ∗ tn − ε n v ) − ( z n + iξ − x ∗ tn − ε n v ) (cid:111) + n (cid:60){ g µn ( z n + iξ ) − g µn ( z n ) } dξ ≤ ∞ (cid:90) e − n tn ξ + n (cid:82) log | iξzn − s | dµ n ( s ) dξ. Using the fact that | z n − s | ≥ √ t n andlog | ix | = 12 log(1 + x ) ≤ x , x ∈ R , we obtain I (1) n ≤ √ t n (cid:90) ∞ e n (cid:16) − x + log(1+ x ) (cid:17) dx ≤ √ t n (cid:90) ∞ e (cid:16) − x + log(1+ x ) (cid:17) dx, and this integral is convergent. (cid:3) Combining (5.6) with Lemma 5.4 and Lemma 5.5, we obtain I n ( u, v ) = O (cid:18) nε n | γ ± n |√ t n δ n e − n tn L n (cid:19) , as n → ∞ . Using also Lemma 2.1, the definition of L n , and the fact that ε n = o ( δ n ), as n → ∞ , we get I n ( u, v ) = O (cid:18) n ε n δ n e − n tn L n (cid:19) = O (cid:16) n e − n n (cid:17) , as n → ∞ , and this yields (5.5). ppendix A. Exact expression for the correlation ker-nel K n,t It is the purpose of this Appendix to derive the explicit representation of the correlationkernel K n,t , on which we based our proofs. Let us first assume that a ( n )1 , . . . , a ( n ) n are notdeterministic but random points following a polynomial ensemble of the form1 Z n ∆ n ( a ( n ) ) det (cid:104) f k − (cid:16) a ( n ) j (cid:17)(cid:105) nj,k =1 (A.1)for certain functions f , . . . , f n − , where Z n > n ( a ( n ) )denotes the Vandermonde determinant∆ n ( a ( n ) ) = (cid:89) j Let n ∈ N , ≤ k ≤ n and t > . Then the function ˜ K n,t ( x, y ) := n πit n (cid:88) k =1 (cid:90) x + i ∞ x − i ∞ dz (cid:89) j (cid:54) = k z − a ( n ) j a ( n ) k − a ( n ) j e n t (( z − x ) − ( a ( n ) k − y ) ) atisfies ρ kn,t ( x , . . . , x k ) = det (cid:16) ˜ K n,t ( x i , x j ) (cid:17) ≤ i,j ≤ k , x , . . . , x k ∈ R , where ρ kn,t denotes the k -th correlation function of P n,t (cf. (1.3) and (1.8) ).Proof. Let us first assume that the points a ( n )1 , . . . , a ( n ) n are pairwise distinct. Then it isknown (see [16, p.15] and references therein) that the eigenvalues y , . . . , y n of M n + √ tH n have a joint probability density function of the form1 C n ∆ n ( y )∆ n ( a ( n ) ) det ( f k − ( y j )) nj,k =1 (A.3)with a normalization constant C n > f k ( y ) = e − n t ( y − a ( n ) k +1 ) , k = 0 , . . . , n − , where a ( n )1 , . . . , a ( n ) n are the deterministic eigenvalues of M n . We show that a kernel ofthis determinantal ensemble is given by˜ K n,t ( x, y ) := n (cid:88) j =1 ˆ p j ( x )ˆ q j ( y ) , where we define ˆ p j ( x ) = n πit x + i ∞ (cid:90) x − i ∞ (cid:96) j,n ( z ) e n t ( z − x ) dz, with (cid:96) j,n ( z ) = (cid:89) ν (cid:54) = j z − a ( n ) ν a ( n ) j − a ( n ) ν , and ˆ q j ( y ) = f j − ( y ) = e − n t ( y − a ( n ) j ) . In order to prove this, we will show that the following biorthogonality relation holds ∞ (cid:90) −∞ ˆ p j ( x )ˆ q k ( x ) dx = δ j,k , j, k ∈ { , . . . , n } . To this end, we make use of the (inverse) Weierstrass transformation of a function ϕ (see[16, p.16] and references) given by W ϕ ( y ) = 1 √ π ∞ (cid:90) −∞ ϕ ( t ) e − ( t − y ) dt, W − Φ( x ) = 1 √ πi i ∞ (cid:90) − i ∞ Φ( s ) e ( s − x ) ds. For a polynomial P and k = 1 , . . . , n we have ∞ (cid:90) −∞ P ( x ) e − n t ( x − a ( n ) k ) dx = (cid:114) tn ∞ (cid:90) −∞ P (cid:32)(cid:114) tn x (cid:33) e − (cid:16) x − √ nt a ( n ) k (cid:17) dx = (cid:114) πtn W ˜ P (cid:18)(cid:114) nt a ( n ) k (cid:19) , (A.4) here W acts on ˜ P defined by ˜ P ( x ) = P (cid:32)(cid:114) tn x (cid:33) . Moreover, letting x = 0 and after a linear change of variables in the definition of ˆ p j weobtainˆ p j ( x ) = 12 πi (cid:114) nt i ∞ (cid:90) − i ∞ (cid:96) j,n (cid:32)(cid:114) tn s (cid:33) e ( s − √ nt x ) ds = (cid:114) n πt W − ˜ (cid:96) j,n (cid:18)(cid:114) nt x (cid:19) , (A.5)where W − acts on ˜ (cid:96) j,n defined by˜ (cid:96) j,n ( x ) = (cid:96) j,n (cid:32)(cid:114) tn x (cid:33) . Using (A.5) and the definition of ˆ q k we get ∞ (cid:90) −∞ ˆ p j ( x )ˆ q k ( x ) dx = ∞ (cid:90) −∞ (cid:114) n πt W − ˜ (cid:96) j,n (cid:18)(cid:114) nt x (cid:19) e − n t ( x − a ( n ) k ) dx. Now, applying (A.4) to the polynomial P ( x ) = (cid:114) n πt W − ˜ (cid:96) j,n (cid:18)(cid:114) nt x (cid:19) gives ∞ (cid:90) −∞ ˆ p j ( x )ˆ q k ( x ) dx = (cid:114) πtn W ˜ P (cid:18)(cid:114) nt a ( n ) k (cid:19) = (cid:96) j,n ( a ( n ) k ) = δ j,k , j, k ∈ { , . . . , n } . Finally, we observe that the joint probability function in (A.3) as well as the kernel ˜ K n,t both depend continuously on the initial points a ( n )1 , . . . , a ( n ) n , so by a continuity argumentthe statement follows for arbitrary initial configurations. (cid:3) We can also write the sum as a contour integral: if γ is a contour encircling all a ( n ) j ’sin the positive sense and if x is such that γ and x + i R do not intersect, we can write˜ K n,t as a double contour integral,˜ K n,t ( x, y ) = n (2 πi ) t (cid:90) x + i ∞ x − i ∞ dz (cid:90) γ dw z − w (cid:81) nj =1 ( z − a ( n ) j ) (cid:81) nj =1 ( w − a ( n ) j ) e n t ( z − x ) e n t ( w − y ) ) , which follows from a residue calculation. If γ and x are such that x + i R has exactlytwo intersection points τ = x + is and τ = x + is with γ , with (cid:61) τ > > (cid:61) τ andthe line segment [ τ , τ ] is fully contained in γ , we have˜ K n,t ( x, y ) = n (2 πi ) t (cid:90) x + i ∞ x − i ∞ dz (cid:90) γ dw z − w (cid:81) nj =1 ( z − a ( n ) j ) (cid:81) nj =1 ( w − a ( n ) j ) e n t ( z − x ) e n t ( w − y ) ) + n πit (cid:90) τ τ dz e n t ( z − x ) e n t ( z − y ) ) . f we take τ and τ as complex conjugates, i.e., s = − s = s > 0, we obtain uponevaluating the second term,˜ K n,t ( x, y ) = n (2 πi ) t (cid:90) x + i ∞ x − i ∞ dz (cid:90) γ dw z − w (cid:81) nj =1 ( z − a ( n ) j ) (cid:81) nj =1 ( w − a ( n ) j ) e n t ( z − x ) e n t ( w − y ) ) + 1 π ( x − y ) sin (cid:16) ( x − y ) s nt (cid:17) e n t ( x − y ) e − ( x − y ) x nt . (A.6)Let g be defined as g µ ( z ) = (cid:90) log( z − a ) dµ ( a ) , for any compactly supported probability measure µ on R . The function g µ depends ingeneral on the choice of the logarithm, if the support of µ is contained in an interval I ⊂ R , then e ng µ ( z ) is independent of the choice of logarithm for z outside I .With this notion we can rewrite (A.6) as˜ K n,t ( x, y ) = n (2 πi ) t (cid:90) x + i ∞ x − i ∞ dz (cid:90) γ dw z − w e n t ( ( z − x ) +2 tg µn ( z ) ) e n t (( w − y ) +2 tg µn ( w )) + 1 π ( x − y ) sin (cid:16) ( x − y ) s nt (cid:17) e n t ( x − y ) e − ( x − y ) x nt . Now, ˜ K n,t and K n,t from (1.9) satisfy the relation˜ K n,t = K n,t f ( x ) f ( y ) , f ( x ) = e n t ( x − xx ) , which means that they define the same determinantal point process. Acknowledgements. 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