Rank and Invertibility of Dense Signed Random Regular Matrices

Let $A$ be the (non-symmetric) adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices, and let $\mathrm{\Xi }$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min (d,n-d)\ge \lambda n$ for some fixed $\lambda \in (0,1\mathrm{/}2]$. We show that there exists $c>0$, depending only on $\lambda$, such that $${\mathbf{P}}_{A,\mathrm{\Xi }}\{\mathrm{rank}(A\circ \mathrm{\Xi })\le n-k\}\le e^{-cnk},1\le k\le n\mathrm{.}$$ As an ingredient in the proof of the rank bound, we use the case $\kappa =0$ of the following quantitative smallest singular value estimate: $${\mathbf{P}}_{A,\mathrm{\Xi }}\{s_n(A\circ \mathrm{\Xi })\le \kappa \}\le C\kappa \sqrt{n}+e^{-c^{\mathrm{\prime }}n},\kappa \ge 0,$$ where $C,c^{\mathrm{\prime }}>0$ depend only on $\lambda$.