  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
$A_n \cdot (pn)^{-1/2}$ is approximately uniform on the unit disk as
$n\rightarrow \infty$ as long as $pn \rightarrow \infty$, 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 avoids the use of $\epsilon$-nets entirely 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.
