Bulk Eigenvalue Correlation Statistics of Random Biregular Bipartite Graphs

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.