Universality for general Wigner-type matrices

TL;DR

This paper proves that the local eigenvalue distribution of large Wigner-type matrices with general variance profiles converges to a universal form, extending previous results to non-stochastic variance matrices.

Contribution

It establishes a local law and bulk universality for Wigner-type matrices with arbitrary variance profiles, not necessarily stochastic, generalizing prior results.

Findings

Resolvent converges to a diagonal matrix with a specific vector equation.

Proves local laws down to minimal spectral scales.

Demonstrates bulk universality for symmetric and hermitian cases.

Abstract

We consider the local eigenvalue distribution of large self-adjoint $N\times N$ random matrices $\mathbf{H}=\mathbf{H}^*$ with centered independent entries. In contrast to previous works the matrix of variances $s_{ij} = \mathbb{E}\, |h_{ij}|^2 $ is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper [1]. We show that as $N$ grows, the resolvent, $\mathbf{G}(z)=(\mathbf{H}-z)^{-1}$, converges to a diagonal matrix, $ \mathrm{diag}(\mathbf{m}(z)) $, where $\mathbf{m}(z)=(m_1(z),\dots,m_N(z))$ solves the vector equation $ -1/m_i(z) = z + \sum_j s_{ij} m_j(z) $ that has been analyzed in [1]. We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.