Local Marchenko-Pastur Law for Random Bipartite Graphs

This paper is the first chapter of three of the author's undergraduate thesis. We study the random matrix ensemble of covariance matrices arising from random $(d_b, d_w)$-regular bipartite graphs on a set of $M$ black vertices and $N$ white vertices, for $d_b \gg \log^4 N$. We simultaneously prove that the Green's functions of these covariance matrices and the adjacency matrices of the underlying graphs agree with the corresponding limiting law (e.g. Marchenko-Pastur law for covariance matrices) down to the optimal scale. This is an improvement from the previously known mesoscopic results. We obtain eigenvector delocalization for the covariance matrix ensemble as consequence, as well as a weak rigidity estimate.