The circular law for sparse random combinatorial matrices

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 $C$.