The almost sure semicircle law for random band matrices with dependent 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 Hochstättler et al. (2015) by Hochstättler, 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 law 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 in the correlated case. Examples that fit our general setup include Curie–Weiss distributed, correlated Gaussian, and as a special case, independent entries.