The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

TL;DR

This paper advances the understanding of the Wigner-Dyson-Mehta conjecture by proving new cases of eigenvalue correlation convergence for Wigner matrices under weaker assumptions, including finite moments and regularity conditions.

Contribution

It establishes the bulk universality conjecture for Wigner matrices with minimal assumptions, improving convergence results and broadening applicability.

Findings

Proved convergence of eigenvalue correlations under finite moment conditions.

Achieved convergence in the vague sense and upgraded to local $L^1$ convergence with regularity.

Extended previous results to weaker hypotheses on atom distributions.

Abstract

A well known conjecture of Wigner, Dyson, and Mehta asserts that the (appropriately normalized) $k$-point correlation functions of the eigenvalues of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in various senses) to the $k$-point correlation function of the Dyson sine process in the asymptotic limit $n \to \infty$. There has been much recent progress on this conjecture, in particular it has been established under a wide variety of decay, regularity, and moment hypotheses on the underlying atom distribution of the Wigner ensemble, and using various notions of convergence. Building upon these previous results, we establish new instances of this conjecture with weaker hypotheses on the atom distribution and stronger notions of convergence. In particular, assuming only a finite moment condition on the atom distribution, we can obtain convergence in the vague sense, and assuming an additional regularity condition, we can upgrade this convergence to locally $L^1$ convergence.