Invertibility of adjacency matrices for random d-regular directed graphs

Let $d\ge 3$ be a fixed integer, and a prime number $p$ such that $\gcd (p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random matrix in ${\mathbb{F}}_p$, \begin{equation} {\mathbb P}($A\text{ is singular in }{\mathbb{F}}_p$)\leq \frac{1+{$\mathrm{o}$}(1)}{p-1}, \end{equation} as $n$ goes to infinity. As a consequence, as a random matrix in $\mathbb{R}$, \begin{equation} {\mathbb P}($A\text{ is singular in }\mathbb{R}$)={$\mathrm{o}$}(1) \end{equation} as $n$ goes to infinity. This answers an open problem by Frieze [12] and Vu [29,30], for random $d$-regular bipartite graphs. The proof combines a local central limit theorem and a large deviation estimate.