Semicircle Law for a Matrix Ensemble with Dependent Entries

Abstract

We study ensembles of random symmetric matrices whose entries exhibit certain correlations. Examples are distributions of Curie-Weiss-type. We provide a criterion on the correlations ensuring the validity of Wigner's semicircle law for the eigenvalue distribution measure. In case of CurieWeiss distributions this criterion applies above the critical temperature (i. e. β < 1). We also investigate the largest eigenvalue of certain ensembles of Curie-Weiss type and find a transition in its behavior at the critical temperature.

The semicircle law for matrices with ergodic entries

“…If it does not decay to 0 (and have the same sign) the semicircle law cannot be the limit of the ESD. This is good agreement with results on exchangeable processes analysed in Friesen and Löwe (2013a) and Hochstättler et al (2016).…”

“…Similar attempts have been made in Schenker and Schulz-Baldes (2005), where a limited number of correlated entries are admitted, Götze and Tikhomirov (2006), where a martingale structure of the entries is imposed, Friesen and Löwe (2013b), where the diagonals of the (symmetric) matrices were filled with independent Markov chains or in Löwe and Schubert (2016), where the upper triangular part of the matrix is filled with a one-dimensional Markov chain. On the other hand, Friesen and Löwe (2013a) and Hochstättler et al (2016) study matrices where either the diagonals or the entire matrix is built of exchangeable random variables. In particular, in Friesen and Löwe (2013a) it was shown that there is a phase transition: If the correlation of the exchangeable random variables go to 0, the limit of the ESD is the semi-circle law, while otherwise it can be described in terms of a free convolution of the semi-circle law with a limiting law obtained in Bryc et al (2006).…”

Semicircle Law for Generalized Curie–Weiss Matrix Ensembles at Subcritical Temperature

“…The following result is proved in [13]. The purpose of the present paper is to extend this result to subcritical temperatures β > 1.…”

“…Observe that the definition in [13] is even more general than Definition 1. It also allows to replace F β by more general functions F (cf.…”

“…Hochstättler, Kirsch, and Warzel proved in [13] the semicircle law for ensembles of real symmetric matrices where the upper triangular part is filled by what they called approximately uncorrelated random variables [13,Def. 4] (see also Definition 5 below).…”