Spectral Density for Random Matrices with Independent Skew-Diagonals

TL;DR

This paper investigates the eigenvalue distribution of large real symmetric matrices with independent skew-diagonals, revealing that weak correlations lead to Wigner's semi-circle law, while strong correlations produce a free convolution of distributions.

Contribution

It introduces a model with independent skew-diagonals and analyzes how different correlation strengths affect the limiting spectral distribution.

Findings

Weak correlations yield Wigner's semi-circle distribution.

Strong correlations result in a free convolution of distributions.

The study distinguishes between two correlation regimes and their spectral effects.

Abstract

We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same skew-diagonal and we distinguish between two types of such correlations, a rather weak and a rather strong one. For weak correlations the limiting distribution is Wigner's semi-circle distribution; for strong correlations it is the free convolution of the semi-circle distribution and the limiting distribution for random Hankel matrices.