Sharp invertibility of random Bernoulli matrices

Abstract

Let $p\in (0,1/2)$ be fixed, and let $B_n(p)$ be an $n\times n$ random matrix with i.i.d. Bernoulli random variables with mean $p$. We show that for all $t\ge 0$, \[\mathbb{P}[s_n(B_n(p)) \le tn^{-1/2}] \le C_p t + 2n(1-p)^{n} + C_p (1-p-ε_p)^{n},\] where $s_n(B_n(p))$ denotes the least singular value of $B_n(p)$ and $C_p,{\epsilon }_p>0$ are constants depending only on $p$. In particular, \[\mathbb{P}[B_{n}(p) \text{ is singular}] = 2n(1-p)^{n} + C_{p}(1-p-ε_p)^{n},\] which confirms a conjecture of Litvak and Tikhomirov. We also confirm a conjecture of Nguyen by showing that if $Q_n$ is an $n\times n$ random matrix with independent rows that are uniformly distributed on the central slice of $\{0,1{\}}^n$, then \[\mathbb{P}[Q_{n} \text{ is singular}] = (1/2 + o_n(1))^{n}.\] This provides, for the first time, a sharp determination of the logarithm of the probability of singularity in any natural model of random discrete matrices with dependent entries.