Person:

Erdos, Laszlo

We prove the universality of the β-ensembles with convex analytic potentials and for any β>0; that is, we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.

Consider (N × N) Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure (\nu_{ij}) with a subexponential decay. Let (\sigma_{ij}^2) be the variance for the probability measure (\nu_{ij}) with the normalization property that (\sum_i\sigma_{ij}^2 = 1) for all j. Under essentially the only condition that (c\leq N\sigma_{ij}^2 \leq c^{−1}) for some constant (c > 0), we prove that, in the limit (N \rightarrow \infty), the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale (M^{−1}).

Consider (N\times N) hermitian or symmetric random matrices (H) with independent entries, where the distribution of the ((i,j)) matrix element is given by the probability measure (\nu_{ij}) with zero expectation and with variance (\sigma_{ij}^2). We assume that the variances satisfy the normalization condition (\sum_{i} \sigma^2_{ij} = 1) for all (j) and that there is a positive constant (c) such that (c\le N \sigma_{ij}^2 \le c^{-1}). We further assume that the probability distributions (\nu_{ij}) have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of (H) is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order ( (N \eta)^{-1}) where (\eta) is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If (\gamma_j =\gamma_{j,N}) denotes the classical location of the (j)-th eigenvalue under the semicircle law ordered in increasing order, then the (j)-th eigenvalue (\lambda_j) is close to (\gamma_j) in the sense that for some positive constants (C, c) (\mathbb P \Big (\exists , j : ; |\lambda_j-\gamma_j| \ge (\log N)^{C\ log\ log\ N} \Big [ \min \big (, j, N-j+1 , \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{c\ log\ log\ N} \big]}) for (N) large enough. (2) The proof of the Dyson's conjecture which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order (N^{-1}). (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large (N) limit provided that the second moments of the two ensembles are identical.

We consider sample covariance matrices of the form X ∗X, where X is an M × N matrix with independent random entries. We prove the isotropic local MarchenkoPastur law, i.e. we prove that the resolvent (X ∗X − z) −1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity hv,(X ∗X − z) −1wi − hv, wim(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v, w ∈ C N . We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Im z > N −1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

We consider N×N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N−3/4+βN−3/4+β for some positive β>0β>0. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V (N(xi − xj)), where x = (x1,..., xN) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrödinger equation. Suppose that the initial data ψN,0 satisfies the energy condition 〈ψN,0, H k NψN,0 〉 ≤ C k N k for k = 1, 2,.... We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density matrices of ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k = 1 but the factorization of ψN,0 is assumed in a stronger sense.

Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N−ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN≫N2/3pN≫N2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.