The sparse circular law, revisited

Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p=p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n·{\left(pn\right)}^{-1/2}$ is approximately uniform on the unit disk as $n\to \infty$ as long as $pn\to \infty$, 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 $pn$ is bounded. One feature of our proof is that it entirely avoids the use of $\epsilon$-nets and, instead, proceeds by studying the evolution of the singular values of the shifted matrices $A_n-zI$ as we incrementally expose the randomness in the matrix.