The circular law for signed random regular digraphs

We consider a large random matrix of the form $Y=A\odot X$, where $A$ the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor p n\rfloor$ for some fixed $p \in (0,1)$, and $X$ is an $n\times n$ matrix of iid centered Bernoulli signs (here $\odot$ denotes the matrix Hadamard product). We prove that as $n\rightarrow \infty$, the empirical spectral distribution of $\frac{1}{\sqrt{d}}Y$ converges weakly in probability to the uniform measure on the unit disk in the complex plane. A key component of our proof is a lower bound on the least singular value of matrices of the form $A\odot X+B$, with $X$ as above, $B$ deterministic, and $A$ a deterministic 0/1 matrix satisfying certain "quasirandomness" conditions.