Rigidity of eigenvalues of generalized Wigner matrices

Abstract

Consider N × 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 νij with zero expectation and with variance σ 2 ij . We assume that the variances satisfy the normalization condition i σ 2 ij = 1 for all j and that there is a positive constant c such that c ≤ N σ 2 ij ≤ c −1 . We further assume that the probability distributions νij 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 η) −1 where η 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 γj = γj,N denotes the classical location of the j-th eigenvalue under the semicircle law ordered in increasing order, then the j-th eigenvalue λj is close to γj in the sense that for some positive constants C, c(2) The proof of Dyson's conjecture [15] which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N −1 up to logarithmic corrections. (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.

Stability of the matrix Dyson equation and random matrices with correlations

“…Local laws have become a cornerstone in the analysis of spectral properties of large random matrices [4,8,20,23,29,35,37,51]. In its simplest form, a local law considers the normalized trace 1 N Tr G(ζ ) of the resolvent.…”

“…Local laws are the first step of a general three step strategy developed in [24,25,27,29] for proving universality. The second step is to add a tiny independent Gaussian component and prove universality for this slightly deformed model via analyzing the fast convergence of the Dyson Brownian motion (DBM) to local equilibrium.…”

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

“…Below this scale the eigenvalue density remains fluctuating even for large N . In [25] the Green function G(z) and its average m(z) have been used to prove edge universality for generalized Wigner matrices. In [32] we derived a local deformed semicircle law for the deformed ensemble (1.1) under the assumption that µ f c has a square root behavior at the endpoints.…”

“…In a series of papers [20,21,22] Erdős, Schlein and Yau showed that the semicircle law for Wigner matrices also holds down to the optimal scale 1/N , up to logarithmic corrections. In [24] a "fluctuation average lemma" was introduced that yielded optimal bounds on the convergence of m(z) for Wigner matrices in the bulk [24] and up to the edge [25] on scales η ≫ N −1 . Below this scale the eigenvalue density remains fluctuating even for large N .…”

Optimal bounds for convergence of expected spectral distributions to the semi-circular law

“…≤ γ nn , the quantiles of G, i.e. G(γ nj ) = j n , and introduce the notation llog n := log log n. Erdös et al [6], [7] showed, for matrices with elements X jk which have a uniformly sub exponential decay, i.e.…”