On the Limiting Spectral Density of Random Matrices filled with Stochastic Processes

TL;DR

This paper investigates how the spectral density of large symmetric random matrices is affected when their entries are filled with stochastic process data rather than independent entries, revealing conditions under which the semi-circle law holds or fails.

Contribution

It introduces a framework for analyzing spectral densities of matrices filled with dependent stochastic process entries, extending classical random matrix theory results.

Findings

Semi-circle law applies under certain filling conditions.

Dependent entries can alter the limiting spectral distribution.

The semi-circle law does not hold for all fillings.

Abstract

We discuss the limiting spectral density of real symmetric random matrices. Other than in standard random matrix theory the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process, which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.