The circular law for sparse non-Hermitian random matrices

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 circular law conjecture states that the empirical measure of the eigenvalues 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 the dimension of the matrix (say n) increases. In the dense regime, after a long series of partial results, the conjecture was established in a seminal work by Tao and Vu.

For sparse random matrices, such as the matrix with i.i.d.Ber\((p_n)\) entries, where \(p_n \) ↓  \(0\) as \(n\)  → \(\infty\), the approach of Tao and Vu can be carried out only when \(n p_n\) grows at a rate polynomial in \(n\) (that is, \(n p_n \sim\) \(n^{\varepsilon}\) for some \(\varepsilon >0\). Beyond that, due to the presence of a large number of zeros their method breaks down. Another example is the adjacency matrices of \(d_n\)-regular directed random graphs with \(n\) vertices, where \(d_n=o(n)\). Here, there is an additional difficulty of dependencies within the entries. In this talk, I will describe new approaches to handle the sparsity and the dependency thereby yielding the circular law limit for the empirical measure of the eigenvalues of these matrices.

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