The sparse circular law, revisited

Abstract

Abstract Let be an matrix with iid entries distributed as Bernoulli random variables with parameter . Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of is approximately uniform on the unit disk as as long as , which is the natural necessary condition. In this paper, we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when is bounded. One feature of our proof is that it entirely avoids the use of ‐nets and, instead, proceeds by studying the evolution of the singular values of the shifted matrices as we incrementally expose the randomness in the matrix.