Singularity of sparse Bernoulli matrices

Abstract

Let $M_n$ be an $n\times n$ random matrix with independent and identically distributed $\mathrm{Bernoulli}(p)$ entries. We show that there is a universal constant $C\ge 1$ such that, whenever p and n satisfy $Clogn∕n\le p\le C^{-1}$,

where $o_n(1)$ denotes a quantity which converges to zero as $n\to \mathrm{\infty }$. We provide the corresponding upper and lower bounds on the smallest singular value of $M_n$ as well.