Circular law for sparse random regular digraphs

TL;DR

This paper proves that the eigenvalue distribution of large sparse random regular digraph adjacency matrices converges to the circular law, extending understanding of spectral behavior in sparse directed graphs.

Contribution

It establishes the circular law for adjacency matrices of sparse random regular digraphs with degrees growing slowly with the number of vertices, using novel techniques for singular value bounds.

Findings

Empirical spectral distribution converges to the circular law.

Convergence holds for degrees growing as a logarithmic power of n.

Develops new methods for bounding singular values in dependent matrix entries.

Abstract

Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ 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, as long as $d\to\infty$ with $n$, the empirical spectral distribution of 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 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 the matrix entries.