A note on the singularity probability of random directed $d$-regular graphs

TL;DR

This paper improves the bound on the probability that the adjacency matrix of a large random $d$-regular graph is singular, using novel methods involving modular singularity analysis and inverse combinatorial results.

Contribution

It introduces a new approach combining modular analysis and inverse combinatorics to better estimate singularity probabilities of random regular graphs.

Findings

Bound on singularity probability is improved to $n^{-1/3+o(1)}$

Method connects singularity analysis with inverse Kneser's problem

Provides insights into the structure of singular adjacency matrices

Abstract

In this note we show that the singular probability of the adjacency matrix of a random $d$-regular graph on $n$ vertices, where $d$ is fixed and $n \to \infty$, is bounded by $n^{-1/3+o(1)}$. This improves a recent bound by Huang. Our method is based on the study of the singularity problem modulo a prime together with an inverse-type result on the decay of the characteristic function. The latter is related to the inverse Kneser's problem in combinatorics.