Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices H = (h _xy ) in ${d\,\geqslant\,1}$ d ? 1 dimensions. The matrix entries h _xy , indexed by ${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$ x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances ${s_{xy} = \mathbb{E} |h_{xy}|^2}$ s x y = E | h x y | 2 . We assume that s _xy is negligible if |x ? y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if ${W\gg L^{4/5}}$ W ? L 4 / 5 . We also show that the magnitude of the matrix entries ${|{G_{xy}}|^2}$ | G x y | 2 of the resolvent ${G=G(z)=(H-z)^{-1}}$ G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute ${\mathbb{E} |{G_{xy}}|^2}$ E | G x y | 2 . We show that, as ${L\to\infty}$ L → ∞ and ${W\gg L^{4/5}}$ W ? L 4 / 5 , the behaviour of ${\mathbb{E} |G_{xy}|^2}$ E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.     

Wigner Random Matrices with Non-Symmetrically Distributed Entries

We show that the spectral radius of an N× N random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by 2 σ + o(N−6/11+ε), where σ² is the variance of the matrix entries and ε is an arbitrary small positive number. Our bound improves the earlier results by Z. Füredi and J. Komlós (1981), and Van Vu (2005).     

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n×n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let x _k denote eigenvalue number k. Under the condition that both k and n?k tend to infinity as n→∞, we show that x _k is normally distributed in the limit. We also consider the joint limit distribution of eigenvalues $(x_{k_{1}},\ldots,x_{k_{m}})$ from the GOE or GSE where k ₁, n?k _m and k _i+1?k _i , 1≤i≤m?1, tend to infinity with n. The result in each case is an m-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.     

The Tracy–Widom Law for Some Sparse Random Matrices

Consider the random matrix obtained from the adjacency matrix of a random d-regular graph by multiplying every entry by a random sign. The largest eigenvalue converges, after proper scaling, to the Tracy–Widom distribution.     

On the Law of Multiplication of Random Matrices

We recover Voiculescu's results on multiplicative free convolutions of probability measures by different techniques which were already developed by Pastur and Vasilchuk for the law of addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the product of two n×n unitary matrices and the measure of the product of three n×n Hermitian (or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar distributed matrices. We establish the convergence in probability as n to a limiting nonrandom measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting measure.     

On the Law of Addition of Random Matrices

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices A _n and B _n rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix U _n (i.e. A _n +U _n ^* B _n U _n ) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of A _n and B _n is obtained and studied. Received: 27 October 1999/ Accepted: 22 March 2000     

Eigenvector Localization for Random Band Matrices with Power Law Band Width

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width W raised to a power μ remains smaller than the matrix size N. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width W, the estimate μ ≤ 8 holds.     

Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries

We consider an ensemble of Wigner symmetric random matrices A_n={a_ij}, i,j=1, . . . ,n with matrix elements a_ij, being i.i.d. symmetrically distributed random variables We assume that and that for p>18. We prove that the distribution of the k (k=1,2, . . . ) largest (smallest) eigenvalues has a universal limit as n→∞ (the Tracy-Widom distribution).     

Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge

This is a continuation of our earlier paper (Tao and Vu, , 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (, 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.     

Shape Fluctuations and Random Matrices

We study a certain random growth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble (GUE).     

Loop-Erased Walks and Random Matrices

Journal of Statistical Physics - It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There...     

Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices

In this paper, we study the complex Wigner matrices $M_{n}=\frac{1}{\sqrt{n}}W_{n}$ whose eigenvalues are typically in the interval [?2,2]. Let λ ₁≤λ ₂?≤λ _n be the ordered eigenvalues of M _n . Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [?2,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as $\mathcal{A}_{n}[f; u]=\sum_{l=1}^{n}f(\lambda_{l})\mathbf{1}_{\{\lambda_{l}\leq u\}}$ . And the second one is $\mathcal{B}_{n}[f; k]=\sum_{l=1}^{k}f(\lambda_{l})$ with positive integer k=k _n such that k/n→y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_{n}[f; \lfloor nt\rfloor]$ . The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on $\mathcal{A}_{n}[f;u]$ for the real Wigner matrices will also be briefly discussed.     

Density of States for Random Band Matrices

 By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W ² , for W large but fixed. Received: 6 February 2002 / Accepted: 17 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by NSF grant DMS 9729992     

Infinite Random Matrices and Ergodic Measures

We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlation in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of “eigenvalues” of infinite Hermitian matrices distributed according to the corresponding measure. Received: 22 January 2001 / Accepted: 30 May 2001     

Random Matrices and Lyapunov Coefficients Regularity

Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a “cone property” are studied as functions of the matrices entries, as long as they vary without destroying the cone property. The result is applied to stability directions, Lyapunov coefficients and Lyapunov exponents of a class of products of random matrices and to dynamical systems. The results are not new and the method is the main point of this work: it is is based on the classical theory of the Mayer series in Statistical Mechanics of rarefied gases.     

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements H _xy , indexed by \({x,y \in \Lambda \subset \mathbb{Z}^d}\), are independent, uniformly distributed random variables if \({\lvert{x-y}\rvert}\) is less than the band width W, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales \({t\ll W^{d/3}}\). We also show that the localization length of the eigenvectors of H is larger than a factor W ^d/6 times the band width. All results are uniform in the size \({\lvert{\Lambda}\rvert}\) of the matrix.     

Edge Universality for Orthogonal Ensembles of Random Matrices

We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of Shcherbina (Commun. Math. Phys. 285, 957–974, 2009) on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble.     

On Universality for Orthogonal Ensembles of Random Matrices

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.     

Classical Skew Orthogonal Polynomials and Random Matrices

Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.