Universality for 1 d random band matrices

TL;DR

This paper proves that 1D random band matrices exhibit universal Wigner-Dyson spectral statistics in the bulk when the band width is sufficiently large, extending the understanding of universality in random matrix theory.

Contribution

It establishes universality for the second correlation function of 1D random band matrices in the bulk spectrum for large band widths using supersymmetric transfer matrix methods.

Findings

Spectral statistics follow Wigner-Dyson distribution for W >> sqrt(N).

Universality holds in the bulk of the spectrum.

Method extends previous supersymmetric approaches to new matrix ensembles.

Abstract

We consider 1d random Hermitian $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k \in\Lambda=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry's variance $J_{jk}=W^{-1}(\delta_{j,k}+\beta\Delta_{j,k})$ in each block. Considering the limit $W, n\to\infty$, we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as $W\gg \sqrt{N}$, is determined by the Wigner -- Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in [Shcherbina, M., Shcherbina, T.:Universality for 1d random band matrices: sigma-model approximation, J.Stat.Phys. 172, p. 627 -- 664 (2018)]