# The circular law for random regular digraphs

## TL;DR

This paper proves that the eigenvalue distribution of large random regular directed graphs follows the Circular Law, extending understanding of spectral properties in random graph models.

## Contribution

It establishes the Circular Law for the spectral distribution of adjacency matrices of random regular digraphs, a significant extension beyond Erdős–Rényi models.

## Key findings

- Empirical spectral distribution converges to the Circular Law.
- Provides quantitative bounds for the smallest singular value.
- Extends spectral law results to regular directed graphs.

## Abstract

Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_n$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_n$.