Dense random regular digraphs: singularity of the adjacency matrix

TL;DR

This paper proves that large random regular directed graphs have invertible adjacency matrices with high probability, using coupling and discrepancy techniques to handle dependencies.

Contribution

It introduces a novel proof combining switchings and discrepancy properties to establish invertibility of adjacency matrices of random regular digraphs.

Findings

Adjacency matrices are invertible with high probability for large random regular digraphs.

The proof employs a couplings approach and discrepancy properties.

Dependencies between matrix entries are effectively managed in the analysis.

Abstract

Fix $c\in (0,1)$ and let $\Gamma$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $\Gamma$ is invertible with high probability. The proof uses a couplings approach based on the switchings method of McKay and Wormald. We also rely on discrepancy properties for the distribution of edges in $\Gamma$, recently proved by the author, to overcome certain difficulties stemming from the dependencies between the entries of $M$.