Real symmetric random matrices and paths counting

TL;DR

This paper provides exact formulas for the expected traces of powers of real symmetric random matrices with independent entries, offering insights into their spectral density and effects of rescaling in the large matrix limit.

Contribution

It derives exact polynomial expressions for the moments of real symmetric random matrices of arbitrary size, extending understanding of their spectral properties.

Findings

Exact evaluation of $<{ m Tr} S^p>$ for various $p$

Provides polynomial formulas in moments of entries

Facilitates analysis of spectral density and rescaling effects

Abstract

Exact evaluation of $<{\rm Tr} S^p>$ is here performed for real symmetric matrices $S$ of arbitrary order $n$, up to some integer $p$, where the matrix entries are independent identically distributed random variables, with an arbitrary probability distribution. These expectations are polynomials in the moments of the matrix entries ; they provide useful information on the spectral density of the ensemble in the large $n$ limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large $n$ limit.