Circular law for random discrete matrices of given row sum

TL;DR

This paper proves that certain random matrices with fixed row sums follow the circular law, extending classical results to matrices with dependent entries and fixed row sums.

Contribution

It establishes the circular law for matrices with fixed row sums, a case not covered by previous i.i.d. entry assumptions, using new bounds on the least singular value.

Findings

Empirical spectral distribution converges to the uniform disk for fixed row sum matrices.

Introduces a polynomial estimate on the least singular value for these matrices.

Extends circular law results to dependent entry matrices with fixed row sums.

Abstract

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}.$$ The circular law theorem in random matrix theory asserts that if the entries of $M_n$ are i.i.d. copies of a random variable with mean zero and variance $\sigma^2$, then the empirical spectral distribution of the normalized matrix $\frac{1}{\sigma\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution $\mu_\cir$ over the unit disk as $n$ tends to infinity. In this paper we show that the empirical spectral distribution of the normalized matrix of $M_n$, a random matrix whose rows are independent random $(-1,1)$ vectors of given row-sum $s$ with some fixed integer $s$ satisfying $|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of $M_n$.