On the singularity of adjacency matrices for random regular digraphs

TL;DR

This paper proves that the adjacency matrices of random regular directed graphs are almost surely invertible under certain degree conditions, using coupling and concentration techniques, and extends results to Hadamard products with expansion properties.

Contribution

It introduces a novel coupling method and concentration results to establish invertibility of adjacency matrices of random regular digraphs and related matrices.

Findings

Adjacency matrices are almost surely invertible under specified degree conditions.

The approach applies to Hadamard products with expansion properties.

Invertibility holds for a broad class of structured random matrices.

Abstract

We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The proof makes use of a coupling of random regular digraphs formed by "shuffling" the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges recently obtained by the author (arXiv:1410.5595). We also apply our general approach to prove a.a.s.\ invertibility of Hadamard products $\Sigma\circ \Xi$, where $\Xi$ is a matrix of iid uniform $\pm1$ signs, and $\Sigma$ is a 0/1 matrix whose associated digraph satisfies certain "expansion" properties.