The effective potential of an M-matrix
TTHE EFFECTIVE POTENTIAL OF AN M -MATRIX MARCEL FILOCHE, SVITLANA MAYBORODA, AND TERENCE TAOA bstract . In the presence of a confining potential V , the eigenfunctions of acontinuous Schr¨odinger operator − ∆ + V decay exponentially with the rate gov-erned by the part of V which is above the corresponding eigenvalue; this can bequantified by a method of Agmon. Analogous localization properties can also beestablished for the eigenvectors of a discrete Schr¨odinger matrix.This note shows, perhaps surprisingly, that one can replace a discrete Schr¨odingermatrix by any real symmetric Z -matrix and still obtain eigenvector localizationestimates. In the case of a real symmetric non-singular M -matrix A (which is asituation that arises in several contexts, including random matrix theory and sta-tistical physics), the landscape function u = A − ff ective po-tential of localization. Starting from this potential, one can create an Agmon-typedistance function governing the exponential decay of the eigenfunctions awayfrom the “wells” of the potential, with each eigenfunction being typically local-ized to a single such well.
1. I ntroduction : history and motivation The fundamental premises of quantum physics guarantee that a potential V in-duces exponential decay of the eigenfunctions of the Schr¨odinger operator − ∆ + V (on either a continuous domain R d or a discrete lattice Z d ) as long as V is largerthan the eigenvalue E outside of some compact region. This heuristic principle hasbeen established with mathematical rigor by S. Agmon [A] and has served as afoundation to many beautiful results in the semiclassical analysis and other fields(see, e.g., [H, HS, S1, S2] for a glimpse of some of them). Roughly speaking, themodern interpretation of this principle is that the eigenfunctions decay exponen-tially away from the “wells” { x : V ( x ) ≤ E } .In 2012, two of the authors of the present paper introduced the concept of the localization landscape . They observed in [FM] that the solution u to the equation M arcel F iloche , L aboratoire de P hysique de la M ati ` ere C ondens ´ ee , E cole P olytechnique ,CNRS, IP P aris , 91128, P alaiseau , F rance S vitlana M ayboroda , S chool of M athematics , U niversity of M innesota , M inneapolis , M in - nesota , 55455, USAT erence T ao , D epartment of M athematics , UCLA, L os A ngeles CA 90095-1555, USAM. Filoche is supported by the Simons foundation grant 601944. S. Mayboroda was partlysupported by the NSF RAISE-TAQS grant DMS-1839077 and the Simons foundation grant 563916.T. Tao is supported by NSF grant DMS-1764034 and by a Simons Investigator Award. a r X i v : . [ m a t h - ph ] J a n THE EFFECTIVE POTENTIAL OF AN M -MATRIX ( − ∆ + V ) u = V and to describe a precise pictureof their exponential decay. For instance, if V takes the values 0 and 1 randomly on atwo-dimensional lattice Z (a classical setting of the Anderson–Bernoulli localiza-tion) the eigenfunctions at the bottom of the spectrum are exponentially localized,that is, exponentially decaying away from some small region, but this would notbe detected by the Agmon theory because the region { V ≤ E } could be completelypercolating and there is no “room” for the Agmon-type decay, especially if theprobability of V = V =
1. And indeed, the phe-nomenon of Anderson localization is governed by completely di ff erent principles,relying on the interferential rather than confining impact of V . On the other hand,looking at the landscape in this example, we observe that the region { u ≤ E } ex-hibits clearly defined isolated wells and that the eigenmodes decay exponentiallyaway from these wells. It turns out that indeed, the reciprocal of the landscape, u ,plays a role of the e ff ective potential , and in [ADFJM1] Arnold, David, Jerison,and the first two authors have proved that the eigenfunctions of − ∆ + V decay ex-ponentially in the regions where { u > E } with the rate controlled by the so-called Agmon distance associated to the landscape, a geodesic distance in the manifolddetermined by ( u − E ) + . The numerical experiments in [ADFJM2] and physicalconsiderations in [ADFJM3] show an astonishing precision of the emerging esti-mates, although mathematically speaking in order to use these results for factualdisordered potentials one has to face, yet again, a highly non-trivial question ofresonances – see the discussion in [ADFJM1]. At this point we have only success-fully treated Anderson potentials via the localization landscape in the context of aslightly di ff erent question about the integrated density of states [DFM].However, the scope of the landscape theory is not restricted to the setting ofdisordered potentials. In fact, all results connecting the eigenfunctions to the land-scape are purely deterministic, and one of the key benefits of this approach is theabsence of a priori assumptions on the potential V , which already in [ADFJM1]allowed us to rigorously treat any operator − div A ∇ + V with an elliptic matrixof bounded measurable coe ffi cients A and any non-negative bounded potential V ,a level of the generality not accessible within the classical Agmon theory. Theseideas and results have been extended to quantum graphs [HM], to the tight-bindingmodel [WZ], and perhaps most notably, to many-body localization in [BLG].This paper shows that the applicability of the landscape theory in fact extendswell beyond the scope of the Schr¨odinger operator, or, for that matter, even thescope of PDEs. Indeed, let us now consider a general real symmetric positivedefinite N × N matrix A = ( a i j ) i , j ∈ [ N ] , which one can view as a self-adjoint operatoron the Hilbert space (cid:96) ([ N ]) on the domain [ N ] (cid:66) { , . . . , N } . In certain situationsone expects A to exhibit “localization” in the following two related aspects, whichwe describe informally as follows: HE EFFECTIVE POTENTIAL OF AN M -MATRIX 3 (i) (Eigenvector localization) Each eigenvector φ = ( φ k ) k ∈ [ N ] of A is local-ized to some index i of [ N ], so that | φ k | decays when | k − i | exceeds somelocalization length L (cid:28) N .(ii) (Poisson statistics) The local statistics of eigenvalues λ , . . . , λ N of A asymp-totically converge to a Poisson point process in a suitably rescaled limit as N → ∞ .Empirically, the phenomena (i) and (ii) are observed to occur in the same matrixensembles A ; intuitively, the eigenvector localization property (i) implies that A “morally behaves like” a block-diagonal matrix, with the di ff erent blocks of A sup-plying “independent” sets of eigenvalues, thus leading to the Poisson statistics in(ii). However, the two properties (i), (ii) are not formally equivalent; for instance,conjugating A by a generic unitary matrix will most likely destroy property (i)without a ff ecting property (ii). Example . Consider the random band matrix Gaussianmodels A , in which the entries a i j are independent Gaussians for 1 ≤ i ≤ j ≤ N and | i − j | ≤ W , but vanish for | i − j | > W , for some 1 ≤ W ≤ N . We refer thereader to [B, §2.2] for a recent survey of this model. If the matrix is normalized tohave eigenvalues E concentrated in the interval [ − ,
2] (and expected to obey theWigner semicircular law π (4 − E ) / + for the asymptotic density of states), it isconjectured (see e.g., [FM2]) that the localization length L should be given by theformula L ∼ min( W (4 − E ) , N );in particular, in the bulk of the spectrum, it is conjectured that localization (inboth senses (i), (ii)) should hold when W (cid:28) N / (with localization length L ∼ W ) and fail when W (cid:29) N / , while near the edge of the spectrum (in which4 − E = O ( L − / )) localization is expected to hold when W (cid:28) N / and fail when W (cid:29) N / . Towards this conjecture, it is known [BYY] in the bulk 4 − E ∼ W (cid:29) N / + ε for any fixed ε >
0, while localizationin sense (i) was established for W (cid:28) N / in [PSSS] (see also [Sh]). In the edge4 − E = O ( L − / ), both directions of the conjecture have been verified in sense (ii)in [S], but the conjecture in sense (i) remains open. Finally, we remark that in theregime W = O (1) the classical theory of Anderson localization [A2] can be used toestablish both (i) and (ii).We now focus on the question of establishing eigenvector localization (i). Canone deduce any uniform bound on the eigenvectors of a general matrix A depicting,in particular, the structure of the exponential decay similarly to the aforementioned One can also study the closely related phenomenon of localization of Green’s functions ( A − z ) − .This latter type of localization is also related to the spectrum of associated infinite-dimensionaloperators consisting of pure point spectrum, thanks to such tools as the Simons–Wol ff criterion[SW]. THE EFFECTIVE POTENTIAL OF AN M -MATRIX considerations for a matrix of the Schr¨odinger operator − ∆ + V ? An immediate ob-jection is that there is no “potential” that could play the role of V . Even aside fromthe fact that the proof of the Agmon decay relies on the presence of both kinetic andpotential energy, as well as on many PDE arguments, it is not clear whether there isa meaningful function, analogous to V , which governs the behavior of eigenvectorsof a general matrix. The main result of this paper is that, perhaps quite surprisingly,the landscape theory still works, at least in the class of real symmetric Z -matrices(matrices with non-positive entries o ff the diagonal). Furthermore, when A is a realsymmetric non-singular M -matrix (a positive semi-definite Z -matrix), the recipro-cal u of the solution to Au = ρ on the index set[ N ] which predicts the exponential decay of the eigenvectors.To be more precise, let us turn to the exact statements.2. M ain results We introduce an Agmon-type distance ρ on the index set [ N ] (cid:66) { , . . . , N } as-sociated to an N × N matrix A , a N × u , and an additionalspectral parameter E ∈ R : Definition 2.1 (Distance) . Let A = ( a i j ) i , j ∈ [ N ] be a real symmetric N × N matrix, let u = ( u i ) i ∈ [ N ] be a vector with all entries non-zero, and let E be a real number. Wedefine the e ff ective potential V = ( v i ) i ∈ [ N ] by the formula(2.2) v i (cid:66) ( Au ) i u i , the shifted e ff ective potential by the formula(2.3) v i (cid:66) ( v i − E ) + (where x + (cid:66) max( x , potential well set by the formula K E (cid:66) { i ∈ [ N ] : v i = } = (cid:26) i ∈ [ N ] : ( Au ) i u i ≤ E (cid:27) , and the distance function ρ = ρ A , u , E : [ N ] × [ N ] → [0 , + ∞ ] by the formula ρ ( i , j ) (cid:66) inf L ≥ inf i ,..., i L ∈ [ N ]: i = i , i L = j (cid:32) L (cid:88) (cid:96) = ln (cid:32) + (cid:115) √ v i (cid:96) v i (cid:96) + | a i (cid:96) i (cid:96) + | (cid:33)(cid:33) where we restrict the infimum to those paths i , . . . , i L for which a i (cid:96) i (cid:96) + (cid:44) (cid:96) = , . . . , L −
1. To put it another way, ρ is the largest pseudo-metric such that(2.4) ρ ( i , j ) ≤ ln (cid:32) + (cid:115) √ v i v j | a i j | (cid:33) whenever a i j (cid:44) HE EFFECTIVE POTENTIAL OF AN M -MATRIX 5 For any set M ⊂ [ N ] we denote by ρ ( i , M ) (cid:66) inf j ∈ M ρ ( i , j ) the distance from agiven index i to M using the distance ρ (with the convention that ρ ( i , M ) = ∞ if M is empty). Similarly, for any set K ⊂ [ N ], we define ρ ( K , M ) (cid:66) inf i ∈ K ρ ( i , M ) forthe separation between K and M .It is easy to see that ρ is a pseudo-metric in the sense that it is symmetric andobeys the triangle inequality with ρ ( i , i ) =
0, although without further hypotheses on A , u , E it is possible that ρ ( i , j ) could be zero or infinite for some i (cid:44) j . One canview ρ as a weighted graph metric on the graph with vertices [ N ] and edges givenby those pairs ( i , j ) with a i j (cid:44)
0, and with weights given by the right-hand side of(2.4).We recall that a
Z-matrix is any N × N matrix A such that a i j ≤ i (cid:44) j , anda M-matrix is a Z -matrix with all eigenvalues having non-negative real part. Ourtypical set-up is the case when A is a real symmetric non-singular M -matrix, i.e., apositive definite matrix with non-positive o ff -diagonal entries, and in that case wewill choose u as the landscape function, i.e., the solution to Au =
1, with 1 denotinga vector with all values equal to 1. We say that a matrix A has connectivity at most W c if every row and column has at most W c non-zero non-diagonal entries. If A isa real symmetric non-singular M -matrix, all the principal minors are positive (seee.g., [P]), and hence by Cram´er’s rule all coe ffi cients of the landscape u will benon-negative. In this case, a simple form of our main results is as follows. Theorem 2.5 (Exponential localization using landscape function) . Let A be a sym-metric N × N M-matrix with connectivity at most W c for some W c ≥ . Let u (cid:66) A − be the landscape function. Assume that ϕ is an (cid:96) -normalized eigenvector of A cor-responding to the eigenvalue E. Let ρ = ρ A , u , E , K = K E be defined by Definition2.1. Then (cid:88) k ϕ k e ρ ( k , K ) √ Wc (cid:18) u k − E (cid:19) + ≤ W c max ≤ i , j ≤ N | a i j | . Informally, the above inequality assures that an eigenvector ϕ experiences expo-nential decay away from the wells of the e ff ective potential V = ( u k ) k ∈ [ N ] cut o ff by the energy level E . This is what typically happens for the Schr ¨odinger oper-ator − ∆ + V (according to some version of the Agmon theory); see for instance[ADFJM1, Corollary 4.5]. However, the existence of such an e ff ective potentialfor an arbitrary M -matrix is perhaps surprising.In fact our results apply to the larger class of real symmetric Z -matrices A andmore general vectors u , and can handle “local” eigenvectors as well as “global”ones. We first introduce some more notation. For instance, if we assume that A is irreducible in the sense that it cannot be expressed (afterpermuting indices) as a block-diagonal matrix, then ρ ( i , j ) will always be finite. THE EFFECTIVE POTENTIAL OF AN M -MATRIX Definition 2.6 (Local eigenvectors) . Let M ⊂ [ N ]. We use I M to denote the N × N diagonal matrix with ( I M ) ii equal to 1 when i ∈ M and 0 otherwise. If ϕ ∈ (cid:96) ([ N ]),we write ϕ | M (cid:66) I M ϕ for the restriction of ϕ to M (extending by zero outside of M ),and similarly if A is an N × N matrix we write A | M (cid:66) I M AI M for the restriction of A to M × M (again extending by zero). We say that a vector ϕ ∈ (cid:96) ([ N ]) is a localeigenvector of A on the domain M with eigenvalue E if ϕ = ϕ | M is an eigenvectorof A | M with eigenvalue E , thus I M ϕ = ϕ and I M AI M ϕ = E ϕ .To avoid confusion we shall sometimes refer to the original notion of an eigen-vector as a global eigenvector ; this is the special case of a local eigenvector inwhich M = [ N ].We can now state a more general form of Theorem 2.5. Theorem 2.7 (Exponential localization) . Let A be a symmetric N × N Z-matrixwith connectivity at most W c for some W c ≥ , and let u be some n × vector ofnon-negative coe ffi cients. Let E > be an energy threshold, and let ρ = ρ A , u , E , v i ,and K E be defined by Definition 2.1. Then for any subset D of [ N ] and any localeigenvector ϕ of A of eigenvalue E ≤ E on D c = [ N ] \ D, one has (2.8) ( E − E ) (cid:88) k (cid:60) K E | ϕ k | e αρ ( k , K E \ D ) + (cid:18) − α W c (cid:19) (cid:88) k (cid:60) K E | ϕ k | e αρ ( k , K E \ D ) v k ≤ W c (cid:107) ϕ (cid:107) max i ∈ K E \ D , j (cid:60) K E \ D | a i j | , for any < α ≤ √ / W c . (Here and in the sequel we use (cid:107) · (cid:107) to denote the (cid:96) ([ N ]) norm.)In particular, if α = √ / W c , E = E, D = ∅ , and ϕ is an (cid:96) -normalized (global)eigenvector of A on the entire domain [ N ] with the eigenvalue E, (2.8) implies that (2.9) (cid:88) k ∈ [ N ] ϕ k e ρ ( k , KE ) √ Wc v k ≤ W c max i , j ∈ [ N ] | a i j | . There are two terms on the left-hand side of (2.8) and they serve di ff erent pur-poses. The bound for the second term (which in particular yields (2.9)) assertsroughly speaking that the eigenvector ϕ k experiences exponential decay in theregime where k is far from K E in the sense that ρ ( k , K E ) (cid:29) √ W c . Note that Theo-rem 2.5 is the special case of (2.9) when A is a M -matrix and u = A − diagonalization , or decoupling , of A on the collection of disjoint subregions defined by the landscape function u , by fol-lowing the arguments from [ADFJM1]. The details are too technical to be put in theintroduction, and we refer the reader to Section 5. In short, the idea is that viewing[ N ] as a graph induced by A (with the vertices connected whenever a i j (cid:44) HE EFFECTIVE POTENTIAL OF AN M -MATRIX 7 define a Voronoi-type splitting of this graph into subgraphs, Ω (cid:96) , each containing anindividual connected component of K E (or sometimes merging a few componentsif convenient). Then A can be essentially decoupled into smaller matrices A | Ω (cid:96) withthe strength of coupling exponentially small in the ρ A , u distance between individ-ual “wells”. Related to this, the spectrum of A will be exponentially close to thecombined spectrum of A | Ω (cid:96) ’s.Note how the geometry of the metric ρ is sensitive to the spatial distribution ofthe matrix A , and in particular to the connectivity properties of the graph induced bythe locations of the nonzero locations of A . For instance, conjugating A by a genericorthogonal matrix will almost certainly destroy the localization of the eigenvectors ϕ , but will also heavily scramble the metric ρ (and most likely also destroy theproperty of being an M -matrix or Z -matrix). On the other hand, conjugating A by apermutation matrix will simply amount to a relabeling of the (pseudo-)metric space([ N ] , ρ ), and not a ff ect the conclusions of Corollary 2.5 and the decoupling resultsin Theorem 5.2 and Corollary 5.5 in any essential way.We will show some results of the numerical simulations in the next section, andthen pass to the proofs, but let us say a few more words about the particular caseswhich would perhaps be of most interest. Random band matrices.
Here the connectivity is W c = W . Strictly speak-ing, the random Gaussian band matrix models A considered in Example 1.1 do notfall under the scope of Corollary 2.5, because the matrices will not be expectedto have non-positive entries away from the diagonal, nor will they be expected tobe positive definite. However, one can modify the model to achieve these prop-erties (at least with high probability), by replacing the Gaussian distributions bydistributions supported on the negative real axis, and then shifting by a suitablepositive multiple of the identity to ensure positive definiteness with high probabil-ity. These changes will likely alter the semicircle law for the bulk distribution ofeigenvalues, but in the spirit of the universality phenomenon, one may still hopeto see localization of eigenvectors, say in the bulk of the spectrum, as long as thewidth W of the band matrix is small enough (in particular when W (cid:28) N / ). Inthis case Corollary 2.5 entails exponential decay of the eigenvectors governed bythe landscape u and Theorem 5.2 and Corollary 5.5 yield the corresponding diag-onalization of A . Of course, the key question is the behavior of the landscape. Ifthe set K E of wells is localized to a short interval, then this corollary will establishlocalization in the spirit of (i) above; however, if K E is instead the union of severalwidely separated intervals then an eigenvector could in principle experience a res-onance in which non-trivial portions of its (cid:96) energy were distributed amongst twoor more of these intervals. Whether or not this happens is governed to some extentby Theorem 5.2 and Corollary 5.5. These results indicate that the resonances haveto be exponentially strong in the distance between the wells, and our numericalexperiments suggest that such strong resonances are in fact quite rare. THE EFFECTIVE POTENTIAL OF AN M -MATRIX Tight-binding Schr¨odinger operators.
When A is a matrix of the tight-bindingSchr¨odinger operator (a standard discrete Laplacian plus a potential) in a cube in Z d , the connectivity parameter W c is now the number of nearest neighbors, 2 d , andthe size of the matrix is the sidelength of the cube to the power d . If the potentialis non-negative, A is an M -matrix with the entries a i j equal to − i (cid:44) j corresponds to the nearest neighbors in the graph structure induced by Z d , and a ii = d + V i . This particular case has been considered in [WZ] and our results clearlycover it. However, the tight-binding Schr¨odinger is only one of many examples,even when concentrating on applications in physics. We can treat any operatorin the form − div A ∇ + V on any graph structure, provided that the signs of thecoe ffi cients yield an M -matrix. We can also address long range hopping for a verywide class of Hamiltonians. Many-body system and statistical physics.
Much more generally, in statisti-cal physics, the probability distribution over all possible microstates (or the den-sity matrix in the quantum setting) of a given system evolves through elementaryjumps between microstates. This evolution is a Markov process whose transitionmatrix is a Z -matrix which is symmetric up to a multiplication by a diagonal ma-trix. For a micro-reversible evolution, the matrix A is symmetric and is akin to aweighted Laplacian on the high-dimensional indirect graph whose vertices are themicrostates and whose edges are the possible transitions.One essential result of statistical physics is that, under condition of irreducibilityof the transition matrix, the system eventually reaches thermodynamical equilib-rium. Our approach might open the way to unravel the structure of the eigenvec-tors of the Markov flow, and thus to understand how localization of these eigen-vectors can induce a many-body system to remain “frozen” for mesoscopic timesout of equilibrium. This e ff ect is referred to as many-body localization . A first suc-cessful implementation of the landscape theory in this context has been recentlyachieved by V. Galitski and collaborators [BLG] for a many-body system of spinswith nearest-neighbor interaction. In this work, the authors cleverly use the ideasof [BAA] to transfer the problem to the Fock space and to deduce an Agmon-typedecay governed by the corresponding e ff ective potential. Once in the Fock space,their results are also a particular case of Theorem 2.7 and Theorem 5.2. From thatpoint, however, the authors of [BLG] go much farther to discuss, based on physi-cal considerations, deep implications of such an exponential decay on many-bodylocalization, but in the present paper we restrict ourselves to mathematics and willnot enter those dangerous waters.Finally, we would like to mention that an idea of trying the localization landscapeand similar concepts in the generality of random matrices has appeared before,e.g., in [LSt] and [GPO + ]. However, the authors relied on a di ff erent principle,extending the inequality | ϕ | ≤ Eu from [FM] to these more general contexts, whichby itself, of course, does not prove exponential decay. The paper [LSt] actually HE EFFECTIVE POTENTIAL OF AN M -MATRIX 9 deals with a di ff erent proxy for the landscape and di ff erent inequalities, but we (andthe authors) believe that these are related to the landscape and that, again, they donot prove exponential decay estimates. However, we would like to mention that theimportance of M -matrices was already suggested in [GPO + ], and it was inspiringand reassuring to arrive at the same setting from such di ff erent points of view.3. N umerical simulations We ran numerical simulations to compute the localization landscape u , the e ff ec-tive potential u , and the eigenvectors for several realizations of random symmetric M -matrices. The diagonal coe ffi cients are random variables which follow a cen-tered normal law of variance 1. The o ff -diagonal coe ffi cients belonging to the first W c / ff -diagonal coe ffi cientsof the upper triangle are taken to be zero, and the lower triangle is completed bysymmetry. This creates A , a Z -matrix of bandwidth W c + W c ).To ensure positivity, we add a multiple of the identity(3.1) A : = A + a I where a = ε − λ ,λ being the smallest eigenvalue of A and ε = .
1. The smallest eigenvalue ofthe resulting matrix A is thus ε . The matrices A and A clearly have the sameeigenvectors and their spectra di ff er only by a constant shift.Below are the results of several simulations. Figures 1-4 correspond to randomsymmetric M -matrices constructed as above of connectivity W c =
2, 6, 20, and 32.Each figure consists of two frames:The top frame displays the localization landscape u superimposed with the first5 eigenvectors plotted in log scale. The exponential decay of the eigenvectorscan clearly be observed on this frame for W c =
2, 6, and 20. One can see that,as expected, it starts disappearing around W c =
32 ( W c being in this case roughlyequal to √ N ). It is important to observe that in all cases the eigenvectors decayexponentially except for the wells of u (equivalently, the peaks of u ) where theystay flat. This is exactly the prediction of Theorem 2.7.The bottom frame displays the e ff ective potential u superimposed with the first5 eigenvectors plotted in linear scale. The horizontal lines indicate the energies ofthe corresponding eigenvectors. One can clearly see the localization of the eigen-vectors inside the wells of the e ff ective potential. M -MATRIX F igure
1. (Top) Localization landscape (blue line) and the 5 firsteigenvectors (in log scale) for a random 3-band symmetric M -matrix. (Bottom) E ff ective potential ( u ) and the first eigenvectors(in linear scale).F igure
2. (Top) Localization landscape (blue line) and the 5 firsteigenvectors (in log scale) for a random 7-band symmetric M -matrix. (Bottom) E ff ective potential ( u ) and the first eigenvectors(in linear scale). HE EFFECTIVE POTENTIAL OF AN M -MATRIX 11 F igure
3. (Top) Localization landscape (blue line) and the 5 firsteigenvectors (in log scale) for a random 21-band symmetric M -matrix. (Bottom) E ff ective potential ( u ) and the first eigenvectors(in linear scale).F igure
4. (Top) Localization landscape (blue line) and the 5 firsteigenvectors (in log scale) for a random 33-band symmetric M -matrix. (Bottom) E ff ective potential ( u ) and the first eigenvectors(in linear scale). M -MATRIX F igure
5. Scatter plots of the logarithm of the absolute value ofseveral eigenvectors against the corresponding Agmon distance, for3 di ff erent values of the connectivity W c =
2, 6, and 20. For eacheigenvector (eigenvectors − ln | ψ i | at anygiven point i vs. the Agmon distance between the point i and thelocation where | ψ | is maximal. The plots exhibit a strong linear rela-tionship between these two quantities, down to values of | ψ i | around e − (of the order of 10 − ), which is a signature of the exponentialdecay. The slope seems to depend only on W c .Figure 5 provides numerical evidence for finer e ff ects encoded in Theorems 2.7and 5.2. The two Theorems combined prove exponential decay away from thewells of the e ff ective potential governed the Agmon distance associated to 1 / u , atleast in the absence of resonances. In Figure 5 we display, for several values ofconnectivity W c and several eigenvectors, the values − ln | ψ i | against the distance ρ A , u , E ( i , i max ), taking as the origin the point i max where | ψ | is maximal, and using thecorresponding eigenvalue as the threshold E . The linear correspondence down to e − is quite remarkable and shows that e − c ρ A , u , E ( i , i max ) is not only an upper bound,but actually an approximation of the eigenfunction, and that the resonances areindeed unlikely. On the other hand, the constant c does not appear to be equal to1 / √ W c which means that in this respect our analysis is probably not optimal, atleast in the class of random matrices. Indeed, we believe that the application ofthe deterministic Schur test in the proof does not yield the best possible constantfor random coe ffi cients, but since we emphasize the universal deterministic results,this step cannot be further improved. HE EFFECTIVE POTENTIAL OF AN M -MATRIX 13
4. T he proof of the main results
In this section we prove Theorem 2.7. We will use a double commutator method.Let [ A , B ] (cid:66) AB − BA denote the usual commutator of N × N matrices, and (cid:104) , (cid:105) theusual inner product on (cid:96) ([ N ]). We observe the general identity(4.1) (cid:104) [[ A , D ] , D ] u , u (cid:105) = (cid:88) i , j ∈ [ N ]: i (cid:44) j a i j u i u j ( d ii − d j j ) . whenever A = ( a i j ) i , j ∈ [ N ] is a matrix, D = diag( d , . . . , d nn ) is a diagonal matrix,and u = ( u i ) i ∈ [ N ] is a vector. In particular we have(4.2) (cid:104) [[ A , D ] , D ] u , u (cid:105) ≤ A is a Z -matrix and the entries of u have constant sign. It will be thisnegative definiteness property that is key to our arguments. One can compare (4.1),(4.2) to the Schr¨odinger operator identity (cid:104) [[ − ∆ + V , g ] , g ] u , u (cid:105) = − (cid:90) R d |∇ g | | u | ≤ ffi ciently well-behaved) functions V , g , u : R d → R .To exploit (4.1) we will use the following identity. Lemma 4.3 (Double commutator identity) . Let A , Ψ , G be N × N real symmetricmatrices such that Ψ G = G Ψ , and suppose that u is an N × vector. Then (cid:104) G [ Ψ , A ] u , G Ψ u (cid:105) = (cid:104) [[ A , G Ψ ] , G Ψ ] u , u (cid:105) − (cid:104) [[ A , G ] , G ] Ψ u , Ψ u (cid:105) . Proof.
By the symmetric nature of G we have (cid:104) [[ A , G ] , G ] Ψ u , Ψ u (cid:105) = (cid:104) GA Ψ u , G Ψ u (cid:105) − (cid:104) AG Ψ u , G Ψ u (cid:105) and similarly from the symmetric nature of G Ψ we have (cid:104) [[ A , G Ψ ] , G Ψ ] u , u (cid:105) = (cid:104) G Ψ Au , G Ψ u (cid:105) − (cid:104) AG Ψ u , G Ψ u (cid:105) . The claim follows. (cid:3)
We can now conclude
Corollary 4.4.
Let A = ( a i j ) i , j ∈ [ N ] be a N × N real symmetric Z-matrix. Assumethat D is some subset of [ N ] and that ϕ is a local eigenvector of A correspondingto the eigenvalue E on D c = [ N ] \ D. Let u = ( u i ) i ∈ [ N ] be a vector with all positiveentries, and let G = diag( G , . . . , G NN ) be a real diagonal matrix. Then (4.5) (cid:88) k ∈ [ N ] ϕ k G kk (cid:18) ( Au ) k u k − E (cid:19) ≤ − (cid:88) i , j ∈ [ N ]: i (cid:44) j a i j ϕ i ϕ j ( G ii − G j j ) . M -MATRIX Proof.
We apply Lemma 4.3 with Ψ (cid:66) diag( ϕ / u , . . . , ϕ N / u n ) (so that Ψ u = ϕ ),write [ Ψ , A ] = Ψ ( A − EI ) − ( A − EI ) Ψ , and use (4.1), (4.2) to conclude that (cid:104) G Ψ ( A − EI ) u , G Ψ u (cid:105) − (cid:104) G ( A − EI ) Ψ u , G Ψ u (cid:105) ≤ − (cid:88) i , j ∈ [ N ]: i (cid:44) j a i j ϕ i ϕ j ( G ii − G j j ) . Since G is diagonal, and ϕ = Ψ u is a local eigenvector on D c , the second term onthe left-hand side is equal to zero. Indeed, ( Ψ u ) k = ( G Ψ u ) k = k ∈ D , andhence (cid:104) G ( A − EI ) Ψ u , G Ψ u (cid:105) = (cid:104) G ( A − EI ) | D c Ψ u , G Ψ u (cid:105) = . Writing Ψ ( A − EI ) u = (cid:18)(cid:18) ( Au ) k u k − E (cid:19) ϕ k (cid:19) k ∈ [ N ] the claim follows. (cid:3) The strategy is then to apply this corollary with a su ffi ciently slowly varyingfunction G , so that one can hope to mostly control the right-hand side of (4.5) bythe left-hand side. Proof of Theorem 2.7.
We abbreviate K E as K for simplicity. We can of courseassume that ϕ is not identically zero. If K \ D was empty we could apply Corollary4.4 with G kk = K \ D is non-empty. We apply Corollary 4.4 with G ii (cid:66) i (cid:60) K \ D e αρ A , u , E ( i , K \ D ) , where the indicator 1 i (cid:60) K \ D is equal to zero for i ∈ K \ D and equal to 1 otherwise.By construction, G kk vanishes for k ∈ K \ D , and ϕ vanishes on D , so that G kk ϕ k vanishes on K . Thus by (2.3) (cid:0) E − E (cid:1) (cid:88) k (cid:60) K ϕ k e αρ ( i , K ) + (cid:88) k (cid:60) K ϕ k e αρ ( i , K ) v k = (cid:88) k (cid:60) K ϕ k G kk (cid:0) E − E (cid:1) + (cid:88) k (cid:60) K ϕ k G kk (cid:18) ( Au ) k u k − E (cid:19) + = (cid:88) k (cid:60) K ϕ k G kk (cid:0) E − E (cid:1) + (cid:88) k (cid:60) K ϕ k G kk (cid:18) ( Au ) k u k − E (cid:19) = (cid:88) k ∈ [ N ] ϕ k G kk (cid:18) ( Au ) k u k − E (cid:19) ≤ − (cid:88) i , j ∈ [ N ]: a ij (cid:44) i (cid:44) j a i j ϕ i ϕ j ( G ii − G j j ) . (4.6) HE EFFECTIVE POTENTIAL OF AN M -MATRIX 15 Now we need to estimate the quantity G ii − G j j whenever a i j (cid:44)
0. We first observefrom the triangle inequality and (2.4) that | e αρ ( i , K \ D ) − e αρ ( j , K \ D ) | ≤ e αρ ( i , K \ D ) (cid:0) e αρ ( i , j ) − (cid:1) ≤ e αρ ( i , K \ D ) (cid:32)(cid:32) + (cid:115) √ v i v j | a i j | (cid:33) α − (cid:33) ≤ e αρ ( i , K \ D ) α (cid:115) √ v i v j | a i j | and similarly with i and j reversed; in particular | e αρ ( i , K \ D ) − e αρ ( j , K \ D ) | ≤ α e αρ ( i , K \ D ) e αρ ( j , K \ D ) √ v i v j | a i j | . Thus we have(4.7) ( G ii − G j j ) ≤ α e αρ ( i , K \ D ) e αρ ( j , K \ D ) √ v i v j | a i j | when i , j (cid:60) K \ D .Next, suppose that i (cid:60) K \ D , j ∈ K \ D . Then G j j =
0, and from (2.4) we have ρ ( i , K \ D ) ≤ ρ ( i , j ) ≤ ln (cid:32) + (cid:115) √ v i v j | a i j | (cid:33) = v j =
0. We conclude that ( G ii − G j j ) = i ∈ K \ D and j (cid:60) K \ D . Finally, if i , j ∈ K \ D then G ii = G j j =
0, so that ( G ii − G j j ) = α (cid:88) i , j (cid:60) K \ D : i (cid:44) j ; a ij (cid:44) | ϕ i || ϕ j | e αρ ( i , K \ D ) e αρ ( j , K \ D ) √ v i v j + (cid:88) i ∈ K \ D ; j (cid:60) K \ D or i (cid:60) K \ D , j ∈ K \ D ; a ij (cid:44) | a i j || ϕ i || ϕ j | . (4.8)Since A has at most W c non-zero non-diagonal entries in each row and column, wesee from Schur’s test (or the Young inequality ab ≤ a + b ) that (cid:88) i , j (cid:60) K \ D : i (cid:44) j ; a ij (cid:44) | ϕ i || ϕ j | e αρ ( i , K \ D ) e αρ ( j , K \ D ) √ v i v j | a i j | ≤ W c (cid:88) i (cid:60) K \ D | ϕ i | e αρ ( i , K \ D ) v i M -MATRIX and (cid:88) i ∈ K \ D ; j (cid:60) K \ D or i (cid:60) K \ D , j ∈ K \ D ; a ij (cid:44) | a i j || ϕ i || ϕ j | ≤ W c ( sup i ∈ K \ D ; j (cid:60) K \ D | a i j | ) (cid:88) i ∈ [ N ] | ϕ i | . Combining all of the above considerations, we arrive at the conclusion of the theo-rem. (cid:3)
5. D iagonalization
Let the notation and hypotheses be as in Theorem 2.7. We abbreviate ρ = ρ A , u , E and K = K E . To illustrate the decoupling phenomenon we place the followinghypothesis on the potential well set K : Hypothesis 5.1 (Separation hypothesis) . There exists a parameter S > , a parti-tion K = (cid:83) (cid:96) K (cid:96) of K into disjoint “wells” K (cid:96) , and “neighborhoods” Ω (cid:96) ⊃ K (cid:96) ofeach well K (cid:96) obeying the following axioms: (i) The neighborhoods Ω (cid:96) are all disjoint. (ii) The neighborhoods Ω (cid:96) contain the S -neighborhood of K (cid:96) , thus ρ ( Ω c (cid:96) , K (cid:96) ) ≥ S . (iii)
For any (cid:96) , we have ρ ( ∂ − Ω (cid:96) , K (cid:96) ) ≥ S , where the inner boundary ∂ − Ω (cid:96) isdefined as the set of all k ∈ Ω (cid:96) such that a k j (cid:44) for some j (cid:60) Ω (cid:96) . We remark that axioms (i), (ii), (iii) imply that the full boundary ∂ Ω (cid:96) , defined asthe union of the inner boundary ∂ − Ω (cid:96) and the outer boundary ∂ + Ω (cid:96) consisting ofthose j (cid:60) Ω (cid:96) such that a k j (cid:44) k ∈ Ω (cid:96) , stays at a distance at S from K ,since every element of an outer boundary ∂ + Ω (cid:96) either lies in the inner boundary ofanother Ω (cid:96) (cid:48) , or else lies outside of all of the Ω (cid:96) (cid:48) . We also remark that axiom (iii) isa strengthening of axiom (ii), since if there was an element k in Ω c (cid:96) at distance lessthan S from K (cid:96) then by taking a geodesic path from K (cid:96) to k one would eventuallyencounter a counterexample to (iii), but we choose to retain explicit mention ofaxiom (ii) to facilitate the discussion below.Informally, to obey Hypothesis 5.1, one should first partition K into “connectedcomponents” K (cid:96) , concatenating two such components together if their separation ρ is too small, so that the separation S (cid:66) inf (cid:96) (cid:44) (cid:96) (cid:48) ρ ( K (cid:96) , K (cid:96) (cid:48) ) is large, and then performa Voronoi-type partition in which Ω (cid:96) consists of those k ∈ [ N ] which lie closer to K (cid:96) in the ρ metric than any other K (cid:96) (cid:48) . The axioms (i), (ii) would then be satisfiedfor any S < S / S is large one wouldexpect axiom (iii) to also be obeyed if we reduce S slightly. It seems plausible thatone could weaken the axiom (iii) and still obtain decoupling results comparable tothose presented here, but in this paper we retain this (relatively strong) axiom inorder to illustrate the main ideas.Let ψ j denote the complete system of orthonormal eigenvectors of A on [ N ]with eigenvalues λ j . Let Ψ ( a , b ) denote the orthogonal projection in (cid:96) ([ N ]) onto the HE EFFECTIVE POTENTIAL OF AN M -MATRIX 17 span of eigenvectors ψ j with eigenvalue λ j ∈ ( a , b ). For a fixed (cid:96) let ϕ (cid:96), j denote acomplete orthonormal system of the local eigenvectors of A on Ω (cid:96) with eigenvalues µ (cid:96), j , and let Φ ( a , b ) be the orthogonal projection onto the span of the eigenvectors ϕ (cid:96), j with eigenvalue µ (cid:96), j ∈ ( a , b ), over all (cid:96) and j .The goal of this section is to prove that under the assumption of Hypothesis 5.1, A can be almost decoupled according to (cid:83) (cid:96) Ω (cid:96) , with the coupling exponentiallysmall in S . More precisely, we have the following result, which is an analogue of[ADFJM1, Theorem 5.1] in the M -matrix setting. Theorem 5.2 (Decoupling theorem) . Assume Hypothesis 5.1. Fix δ > and let ϕ be one of the local eigenvectors ϕ (cid:96), j with eigenvalue µ = µ (cid:96), j and µ ≤ E − δ . Then (5.3) (cid:107) ϕ − Ψ ( µ − δ,µ + δ ) ϕ (cid:107) ≤ W c δ max i , j ∈ [ N ] | a i , j | e − S √ Wc (cid:107) ϕ (cid:107) . Conversely, if ψ is one of the global eigenvectors ψ j with eigenvalue λ = λ j ≤ E − δ ,then (5.4) (cid:107) ψ − Φ ( λ − δ,λ + δ ) ψ (cid:107) ≤ W c δ max i , j ∈ [ N ] | a i , j | e − S √ Wc (cid:107) ψ (cid:107) . Proof . We mimic the arguments from [ADFJM1]. Let us consider the residualvector r (cid:66) A ϕ − µϕ = ( A − A | Ω (cid:96) ) ϕ. Note that the expression ( A − A | Ω (cid:96) ) ϕ only depends on the values of ϕ in the boundaryregion ∂ Ω (cid:96) . From Schur’s test one thus has (cid:107) r (cid:107) ≤ W c / max i , j ∈ [ N ] | a i , j | (cid:107) ϕ (cid:107) (cid:96) ( ∂ Ω (cid:96) ) . We apply Theorem 2.7 with E = µ and D = Ω c (cid:96) , so that K \ D = K ∩ Ω (cid:96) = K (cid:96) and ρ ( k , K \ D ) = ρ ( k , K (cid:96) ) ≥ S for all k ∈ ∂ Ω (cid:96) by Hypothesis 5.1, which then yields (cid:107) ϕ (cid:107) (cid:96) ( ∂ Ω (cid:96) ) ≤ e − α S E − µ W c i , j ∈ [ N ] | a i , j | (cid:107) ϕ (cid:107) . Taking α (cid:66) √ / W c and recalling that E − µ > δ , we have (cid:107) r (cid:107) ≤ δ W c i , j ∈ [ N ] | a i , j | e − S √ Wc (cid:107) ϕ (cid:107) . From the spectral theorem one has (cid:107) r (cid:107) ≥ δ (cid:107) ϕ − Ψ ( µ − δ,µ + δ ) ϕ (cid:107) and the claim (5.3) follows.The proof of (5.4) is somewhat analogous. Let us define the residual vector (cid:101) r (cid:66) (cid:88) (cid:96) ( A | Ω (cid:96) − λ I ) ψ | Ω (cid:96) = (cid:88) (cid:96) I Ω (cid:96) [ A , I Ω (cid:96) ] ψ M -MATRIX where the matrices I Ω (cid:96) were defined in Definition 2.6. The values ( I Ω (cid:96) [ A , I Ω (cid:96) ]) ik are only non-zero when a ik (cid:44) i , k ∈ ∂ Ω (cid:96) . In particular, by Hypothesis 5.1,we have ρ ( k , K ) ≥ S , and ˜ r only depends on the values of ψ outside of the S -neighborhood of K . Applying Theorem 2.7 with E = λ and D = ∅ and applyingSchur’s test as before, we conclude that (cid:107) (cid:101) r (cid:107) ≤ δ W c i , j ∈ [ N ] | a i , j | e − S √ Wc (cid:107) ϕ (cid:107) . On the other hand, from the spectral theorem we have (cid:107) (cid:101) r (cid:107) = (cid:88) (cid:96) (cid:107) ( A | Ω (cid:96) − λ I ) ψ Ω (cid:96) (cid:107) ≥ δ (cid:107) ψ − Φ ( λ − δ,λ + δ ) ψ (cid:107) and the claim (5.4) follows. (cid:3) The theorem above assures that A can be essentially decoupled on the union of Ω (cid:96) ’s in the sense that the eigenvectors of A are exponentially close to the span ofthe eigenvectors of A | Ω (cid:96) , and vice versa. A direct corollary of this result is thatthe eigenvalues of A are also exponentially close to the combined spectrum of A | Ω (cid:96) over all (cid:96) : Corollary 5.5.
Assume Hypothesis 5.1. Fix some δ > . Consider the countingfunctions N ( λ ) (cid:66) { λ j : λ j ≤ λ } ; N ( µ ) (cid:66) { µ (cid:96), j : µ (cid:96), j ≤ µ } . Assume that µ ≤ E and choose a natural number ¯ N such thatW c δ max i , j ∈ [ N ] | a i , j | ¯ N < e S √ Wc . Then (5.6) min( ¯ N , N ( µ − δ )) ≤ N ( µ ) and min( ¯ N , N ( µ − δ )) ≤ N ( µ ) . Proof.
Consider the first p unit eigenvectors ψ , . . . , ψ p of A , where p (cid:66) min( ¯ N , N ( µ − δ )). By definition of the counting function, the eigenvalues λ , . . . , λ p of theseeigenvalues are less than µ − δ . Applying the second conclusion of Theorem 5.2,we conclude that (cid:107) ψ k − Φ (0 ,µ ) ψ k (cid:107) ≤ W c δ max i , j ∈ [ N ] | a i , j | e − S √ Wc for k = , . . . , p , Hence, for any nonzero linear combination ψ = (cid:80) pj = α j ψ j , wehave (cid:107) ψ − Φ (0 ,µ ) ψ (cid:107) ≤ (cid:88) j | α j |(cid:107) ψ j − Φ (0 ,µ ) ψ j (cid:107) HE EFFECTIVE POTENTIAL OF AN M -MATRIX 19 ≤ (cid:16) W c δ max i , j ∈ [ N ] | a i , j | e − S √ Wc (cid:17) / (cid:107) ψ (cid:107) p / ≤ (cid:16) W c δ max i , j ∈ [ N ] | a i , j | e − S √ Wc ¯ N (cid:17) / (cid:107) ψ (cid:107) < (cid:107) ψ (cid:107) . It follows that the restriction of Φ (0 ,µ ) to the span of the ψ j , j = , . . . , p , is injective,and hence the rank N ( µ ) of the matrix Φ (0 ,µ ) is at least p . In other words, N ( µ ) ≥ p . This establishes the latter inequality in (5.6); the former one is establishedsimilarly. (cid:3) Acknowledgements.
We are thankful to Guy David for many discussions in thebeginning of this project and to Sasha Sodin for providing the literature and contextwith regard to the edge state localization.
Data Availability Statement.
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