The circular law for random regular digraphs with random edge weights

TL;DR

This paper proves that the eigenvalue distribution of certain large random matrices derived from random regular digraphs with random weights converges to the uniform distribution on the unit disk, extending the circular law.

Contribution

It establishes the circular law for a new class of random matrices formed from random regular digraphs with random edge weights.

Findings

Empirical spectral distribution converges to the uniform measure on the unit disk.

The result holds for matrices with entries having finite 4+eta moments.

Convergence occurs in probability as matrix size tends to infinity.

Abstract

We consider random $n\times n$ matrices of the form $Y_n=\frac1{\sqrt{d}}A_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor p n\rfloor$ 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.