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

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 $n^{-1/3+o\left(1\right)}$. This improves a recent bound by Huang in Huang (2021). Our method is based on the study of the singularity problem modulo a prime developed in Huang (2021) (and also partially in Mészáros, 2021; Nguyen and Wood, 2018), 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.