The limiting spectral law of sparse iid matrices

Let A be an n by n matrix where each entry is independent and takes 1 with probability p, and 0 otherwise. In this talk we will be interested in the limiting law of the distribution of the eigenvalues; that is, the general “shape” that the distribution of eigenvalues of A has in the complex plane, as n tends to infinity. While the study of these distributions goes back to the work of Wigner from the 1950s, many basic questions are still not understood. Indeed, for small p, (namely p = c/n) it is not even known if this limiting shape converges to a deterministic distribution as n tends to infinity. Here I will discuss the recent resolution of this problem. Based on joint work Ashwin Sah and Mehtaab Sawhney.