Adjacency matrices of random digraphs: singularity and anti-concentration

TL;DR

This paper proves the invertibility of adjacency matrices of random d-regular directed graphs with high probability, establishing new anti-concentration properties of such graphs that are of independent interest.

Contribution

It introduces a novel anti-concentration property for d-regular directed graphs and proves the invertibility of their adjacency matrices with high probability.

Findings

Adjacency matrices are invertible with probability at least 1 - C ln^3 d / sqrt(d).

Established a Littlewood-Offord type anti-concentration property for these graphs.

Proved the property even when part of the graph is fixed or 'frozen'.

Abstract

Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $\delta_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $\delta = (\delta_1, \delta_2, ..., \delta_n)\in \{0, 1\}^n$. Then for every $v\in\{0,1\}^n$ the probability that $\delta=v$ is exponentially small. This property holds even if a part of the graph is "frozen".