Rank and Invertibility of Dense Signed Random Regular Matrices

Abstract

Let $A$ be the (non-symmetric) adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices, and let $\Xi$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min (d,n-d)\ge \lambda n$ for some fixed $\lambda \in (0,1/2]$. We show that there exists $c>0$, depending only on $\lambda$, such that \[ \mathbf P_{A,Ξ}\{\operatorname{rank}(A\circΞ)\le n-k\}\le e^{-c nk},\qquad 1\le k\le n. \] As an ingredient in the proof of the rank bound, we use the case $\kappa =0$ of the following quantitative smallest singular value estimate: \[ \mathbf P_{A,Ξ}\{s_n(A\circΞ)\leκ\} \le Cκ\sqrt n+e^{-c' n}, \qquad κ\ge0, \] where $C,c^{\prime }>0$ depend only on $\lambda$.