The circular law for sparse non-Hermitian matrices

For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j}){i,j=1}^n$,where${\xi{i,j}}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{{\delta }_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, we prove that the empirical spectral distribution of $A_n\mathrm{/}\sqrt{n{p}_n}$ converges weakly to the circular law, in probability, for all $p_n$ such that $p_n=\omega ({\log }^{2}n\mathrm{/}n)$. Additionally if $p_n$ satisfies the inequality $n{p}_n>\exp (c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erd\H{o}s-R'{e}nyi graph with edge connectivity probability $p_n$.