The circular law for sparse non-Hermitian matrices.

For a class of sparse random matrices of the form An=(.i,j.i,j)ni,j=1, where {.i,j} are i.i.d. centered sub-Gaussian random variables of unit variance, and {.i,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=.(log2n/n). Additionally if pn satisfies the inequality npn>exp(clogn.....) 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.s.Rényi graph with edge connectivity probability pn.