The local semicircle law for a general class of random matrices

Abstract

We consider a general class of N ×N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij| 2 . As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W N 1−εn with some εn > 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [6,17,19].

Stability of the matrix Dyson equation and random matrices with correlations

“…This directly follows from the uniform boundedness of Im G x x (ζ ) and spectral decomposition of the resolvent (see e.g. [20]). …”

“…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.…”

“…Thus Theorem 2.6 yields the following: In order, to state such estimates for the error matrix we use the notion of stochastic domination, first introduced in [20], that is designed to compare random variables up to N ε -factors on very high probability sets.…”

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

“…Fluctuation averages of the form (6.1) (with n 0 = 1) and more general fluctuation averages have been studied in [18], see also [17], for generalized Wigner ensembles and random band matrices. In [32], these ideas have been applied to the deformed Wigner ensemble, under the assumptions that the limiting eigenvalue distribution has a square root behavior at the spectral edge.…”

“…Our proof relies on the basic strategy of [17]. However, in our setup the diagonal entries of G are not uniformly bounded, which requires several changes to previous arguments.…”