Limiting spectral distribution for Wigner matrices with dependent entries

TL;DR

This paper establishes the existence and explicit description of the limiting spectral distribution for Wigner matrices with dependent Gaussian entries, including special cases where the distribution simplifies to known laws.

Contribution

It extends the spectral distribution analysis to dependent entries and provides explicit moment and transform characterizations, relaxing Gaussian assumptions in some cases.

Findings

Limiting spectral distribution exists under certain covariance conditions.

Explicit moments and Stieltjes transform characterization provided.

Special cases include free convolution and semicircular law.

Abstract

In this article we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability condition. We provide an explicit description of the moments of the limiting measure. We also show that in some special cases the Gaussian assumption can be relaxed. The description of the limiting measure can also be made via its Stieltjes transform which is characterized as the solution of a functional equation. In two special cases, we get a description of the limiting measure - one as a free product convolution of two distributions, and the other one as a dilation of the Wigner semicircular law.