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 $Ξ$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min(d,n-d)\geλn$ for some fixed $λ\in(0,1/2]$. We show that there exists $c > 0$, depending only on $λ$,