The circular law for sparse non-Hermitian matrices

For a class of sparse random matrices of the form Aₙ=(ξi,jδi,j)i,j=1ⁿ, where \ξi,j\ are i.i.d. centered sub-Gaussian random variables of unit variance, and \δi,j\ are i.i.d. Bernoulli random variables taking value $1$ with probability pₙ, we prove that the empirical spectral distribution of Aₙ/√npₙ converges weakly to the circular law, in probability, for all pₙ such that pₙ=ω(log²n/n). Additionally if pₙ satisfies the inequality npₙ>exp(c√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 Erdős–Rényi graph with edge connectivity probability pₙ.