Title: Local Marchenko-Pastur Law for Random Bipartite Graphs,
Abstract: 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.