# Circular law for sparse random regular digraphs

### Alexander E. Litvak

University of Alberta, Edmonton, Canada### Anna Lytova

University of Opole, Poland### Konstantin Tikhomirov

Princeton University, USA### Nicole Tomczak-Jaegermann

University of Alberta, Edmonton, Canada### Pierre Youssef

Université Paris-Diderot, France

## 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.

## Cite this article

Alexander E. Litvak, Anna Lytova, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann, Pierre Youssef, Circular law for sparse random regular digraphs. J. Eur. Math. Soc. 23 (2021), no. 2, pp. 467–501

DOI 10.4171/JEMS/1015