The Almost Sure Semicircle Law for Random Band Matrices with Dependent Entries

TL;DR

This paper proves that the empirical spectral distribution of certain correlated random band matrices converges almost surely to the semicircle law, extending previous probabilistic results to almost sure convergence under specific conditions.

Contribution

It establishes almost sure convergence of spectral distributions for correlated band matrices, extending prior probability-based results and identifying bandwidth growth conditions.

Findings

Almost sure convergence to the semicircle law for correlated band matrices.

Sufficient conditions for convergence in probability and almost surely.

Examples include Curie-Weiss, correlated Gaussian, and independent entries.

Abstract

We analyze the empirical spectral distribution of random periodic band matrices with correlated entries. The correlation structure we study was first introduced in 2015 by Hochst\"attler, Kirsch and Warzel, who named their setup "almost uncorrelated" and showed convergence to the semicircle distribution in probability. We strengthen their results which turn out to be also valid almost surely. Moreover, we extend them to band matrices. Sufficient conditions for convergence to the semicircle distribution both in probability and almost surely are provided. In contrast to convergence in probability, almost sure convergence seems to require a minimal growth rate for the bandwidth. Examples that fit our general setup include Curie-Weiss distributed, correlated Gaussian, and as a special case, independent entries.