Sparse matrices are abundant in statistics, neural network, financial modeling, electrical engineering, and wireless communications. In the regime of sparse non-Hermitian random matrices, I will describe our work that establishes the celebrated circular law conjecture. The conjecture states that the empirical spectral distribution (ESD) of a (properly scaled) matrix with i.i.d.~entries of zero mean and unit variance converges to the uniform measure on the unit disk in the complex plane, as $n$, the dimension of the matrix increases. In the dense regime, after a series of partial results, the conjecture was established in a seminal work by Tao-Vu.

For sparse random matrices, such as the matrix with i.i.d.~Ber$(p_n)$ entries, where $np_n$ grows at a rate sub-polynomial in $n$, the method of Tao-Vu fails due to the presence of a large number of zeros. In case of the adjacency matrix of a $d_n$-regular directed random graph on $n$ vertices, where $d_n=o(n)$ there is an additional difficulty of dependencies within the entries. I will describe new approaches to handle the sparsity and the dependency thereby yielding the circular law limit for the ESDs of these matrices.

This talk is based on joint works with Nicholas Cook, Mark Rudelson, and Ofer Zeitouni.