Bulk Universality for Generalized Wigner Matrices With Few Moments

TL;DR

This paper proves that generalized Wigner matrices with very few moments still exhibit universal spectral behavior similar to GOE, even at nearly the smallest spectral scales.

Contribution

It establishes bulk universality for generalized Wigner matrices with only finite $(2 + ext{small } \varepsilon)$ moments, extending universality results to matrices with minimal moment assumptions.

Findings

Bulk local semicircle law holds at near the smallest spectral scale.

Spectral statistics converge to GOE predictions at fixed energies.

Universality applies despite minimal moment conditions.

Abstract

In this paper we consider $N \times N$ real generalized Wigner matrices whose entries are only assumed to have finite $(2 + \varepsilon)$-th moment for some fixed, but arbitrarily small, $\varepsilon > 0$. We show that the Stieltjes transforms $m_N (z)$ of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when $\eta = \Im (z)$ is almost of order $N^{-1}$. As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian Orthogonal Ensemble (GOE) in the large $N$ limit.