Random matrices, non-backtracking walks, and orthogonal polynomials

TL;DR

This paper explores the connections between random matrix theory, non-backtracking walks on graphs, and orthogonal polynomials, providing a unified perspective on classical spectral laws.

Contribution

It introduces a novel interpretation of key random matrix results using non-backtracking walks and orthogonal polynomials, offering new insights into spectral distributions.

Findings

Reinterpreted Wigner's law through non-backtracking walks

Connected Marchenko-Pastur law to orthogonal polynomials

Provided a unified framework linking graph walks and spectral measures

Abstract

Several well-known results from the random matrix theory, such as Wigner's law and the Marchenko--Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.