Circular law for sparse random regular digraphs

Abstract

Fix a constant C\geq 1 and let d=d(n) satisfy d\leq \mathrm {ln}^{C} n for every large integer n . Denote by A_n the adjacency matrix of a uniform random directed d -regular graph on n vertices. We show that if d\to\infty as n \to \infty , the empirical spectral distribution of the appropriately rescaled matrix A_n converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in a directed d -regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of A_n based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between matrix entries.

Locations

• arXiv (Cornell University) - View - PDF
• Journal of the European Mathematical Society - View