The smallest singular value of dense random regular digraphs

TL;DR

This paper establishes bounds on the smallest singular value of adjacency matrices of dense random regular digraphs, confirming a conjecture and matching bounds known for i.i.d. Bernoulli matrices.

Contribution

It provides the first optimal bounds for the smallest singular value of dense random regular digraph adjacency matrices, confirming a conjecture of Cook.

Findings

Bound on the probability that the smallest singular value is small

Matching bounds with i.i.d. Bernoulli matrices

Confirmation of Cook's conjecture on singularity probability

Abstract

Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geq\lambda n$. We show that for any $\kappa \geq 0$, \[\mathbb{P}[s_n(A)\leq\kappa]\leq C_\lambda\kappa\sqrt{n}+2e^{-c_\lambda n}.\] Up to the constants $C_\lambda, c_\lambda > 0$, our bound matches optimal bounds for $n\times n$ random matrices, each of whose entries is an i.i.d $\text{Ber}(d/n)$ random variable. The special case $\kappa = 0$ of our result confirms a conjecture of Cook regarding the probability of singularity of dense random regular digraphs.