Large corank of dense random regular digraphs

Let $1\le k\le n$ and let $A$ be the adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices. Suppose that $\lambda n\le d\le (1-\lambda )n$ for a fixed $0<\lambda \le 1\mathrm{/}2$. We show that there exists $c_{\lambda }>0$ depending only on $\lambda$ such that $$\mathbb{P}[\mathrm{rank}(A)\le n-k]\le 2e^{-c_{\lambda }kn}\mathrm{.}$$ This gives a large corank extension of the exponential singularity bound of Jain, Sah, and Sawhney.