The Mar\v{c}enko-Pastur law for sparse random bipartite biregular graphs

TL;DR

This paper proves that the spectral distribution of certain bipartite biregular random graphs converges to a symmetrized Marčenko-Pastur law, both globally and locally, extending previous methods in random matrix theory.

Contribution

It establishes the convergence of spectral distributions for bipartite biregular graphs to a Marčenko-Pastur type law, including local convergence results.

Findings

Spectral distribution converges globally to a symmetrized Marčenko-Pastur law.

Local convergence of spectral distribution on small intervals.

Method parallels previous work by Dumitriu and Pal (2012).

Abstract

We prove that the empirical spectral distribution of a (d_L, d_R)-biregular, bipartite random graph, under certain conditions, converges to a symmetrization of the Mar\v{c}enko-Pastur distribution of random matrix theory. This convergence is not only global (on fixed-length intervals) but also local (on intervals of increasingly smaller length). Our method parallels the one used previously by Dumitriu and Pal (2012).