A phase transition for the limiting spectral density of random matrices

TL;DR

This paper investigates how correlations in the entries of symmetric random matrices influence their spectral distribution, revealing a phase transition between the semicircle law and other limiting laws based on correlation strength.

Contribution

It introduces a model with correlated diagonal entries and characterizes the phase transition in the limiting spectral distribution.

Findings

Spectral distribution follows the semicircle law under weak correlations.

Strong correlations lead to a different limiting spectral law.

The results connect to known laws for Toeplitz matrices.

Abstract

We analyze the spectral distribution of symmetric random matrices with correlated entries. While we assume that the diagonals of these random matrices are stochastically independent, the elements of the diagonals are taken to be correlated. Depending on the strength of correlation the limiting spectral distribution is either the famous semicircle law or some other law, related to that derived for Toeplitz matrices by Bryc, Dembo and Jiang (2006).