The circular law for random regular digraphs with random edge weights

Abstract

We consider random $n\times n$ matrices of the form $Y_n=\frac{1}{\sqrt{d}}{A}_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor pn\rfloor$ for some fixed $p\in (0,1)$, and $X_n$ is an $n\times n$ matrix of iid centered random variables with unit variance and finite $4+\eta$-th moment (here $\circ$ denotes the matrix Hadamard product). We show that as $n\to \infty$, the empirical spectral distribution of $Y_n$ converges weakly in probability to the normalized Lebesgue measure on the unit disk.

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