Bulk Eigenvalue Correlation Statistics of Random Biregular Bipartite Graphs

TL;DR

This paper studies the eigenvalue correlation statistics of random biregular bipartite graphs, demonstrating their stability and universality in the bulk spectrum, aligning with classical random matrix ensembles.

Contribution

It proves the stability and universality of bulk eigenvalue correlation statistics for these graphs, connecting them to GOE and Wishart ensembles.

Findings

Bulk eigenvalue correlations are stable for short times.

Universality matches graphs with GOE and Wishart ensembles.

Results hold for a range of degree parameters.

Abstract

This paper is the second chapter of three of the author's undergraduate thesis. In this paper, we consider the random matrix ensemble given by $(d_b, d_w)$-regular graphs on $M$ black vertices and $N$ white vertices, where $d_b \in [N^{\gamma}, N^{2/3 - \gamma}]$ for any $\gamma > 0$. We simultaneously prove that the bulk eigenvalue correlation statistics for both normalized adjacency matrices and their corresponding covariance matrices are stable for short times. Combined with an ergodicity analysis of the Dyson Brownian motion in another paper, this proves universality of bulk eigenvalue correlation statistics, matching normalized adjacency matrices with the GOE and the corresponding covariance matrices with the Gaussian Wishart Ensemble.