Benjamini-Schramm convergence and limiting eigenvalue density of random matrices

TL;DR

This paper explores how Benjamini-Schramm convergence of random graphs can be used to derive the limiting eigenvalue densities of certain random matrix ensembles, connecting graph limits with spectral laws.

Contribution

It demonstrates a novel application of local graph convergence to compute global eigenvalue distributions of beta-Gaussian and beta-Laguerre ensembles.

Findings

Derivation of Wigner semicircle law via graph limits

Derivation of Marchenko-Pastur law via graph limits

Unified framework linking graph convergence and spectral density

Abstract

We review the application of the notion of local convergence on locally finite randomly rooted graphs, known as Benjamini-Schramm convergence, to the calculation of the global eigenvalue density of random matrices from the beta-Gaussian and beta-Laguerre ensembles. By regarding a random matrix as the weighted adjacency matrix of a graph, and choosing the root of such a graph with uniform probability, one can use the Benjamini-Schramm limit to produce the spectral measure of the adjacency operator of the limiting graph. We illustrate how the Wigner semicircle law and the Marchenko-Pastur law are obtained from this machinery.