Let A be an n by n matrix with iid Ber(d/n) entries. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution. The proof involves incrementally exposing the randomness of the underlying matrix and studying the evolution of the singular values. Another component of the proof relies on understanding the structure of near singular vectors via expansion of the underlying bipartite graph. The talk will focus on the simpler model case where we give a substantially simplified proof of the sparse circular law of Rudelson and Tikhomirov.

Joint with Ashwin Sah and Julian Sahasrabudhe.