The circular law for sparse random combinatorial matrices

TL;DR

This paper proves that the spectral distribution of certain sparse random matrices with fixed row sums converges to the circular law, extending understanding of eigenvalue distributions in sparse regimes.

Contribution

It establishes the circular law for sparse combinatorial matrices with fixed row sums and provides quantitative bounds on the smallest singular value in this setting.

Findings

Spectral distribution converges to the circular law for d=o(n).

Provides lower bounds on the smallest singular value of shifted matrices.

Results hold for d in a specified sparse regime.

Abstract

Let $\log^{2+\varepsilon} n \le d \le n/2$ for some fixed $\varepsilon \in (0,1)$, and let $M_n$ be an $n\times n$ random matrix with entries in ${0,1}$, where each row is independently and uniformly sampled from the set of all vectors in ${0,1}^n$ containing exactly $d$ ones. We show that the empirical spectral distribution of the appropriately rescaled matrix $M_n$ converges in probability to the circular law provided that $d=o(n)$. As a crucial element of the proof, we obtain quantitative lower bounds on the smallest singular value of the shifted matrices $M_n-zI_n$ whenever $|z|\le \sqrt d \log\log d$ and $C\log n \le d \le n/2$ for some absolute positive constant $C$.