Singularity of Bernoulli matrices in the sparse regime $pn = O(\log(n))$

Han Huang

Published 2020-09-29Version 1

Consider an $n\times n$ random matrix $A_n$ with i.i.d Bernoulli($p$) entries. In a recent result of Litvak-Tikhomirov, they proved the conjecture $$ \mathbb{P}\{\mbox{$A_n$ is singular}\}=(1+o_n(1)) \mathbb{P}\big\{\mbox{either a row or a column of $A_n$ equals zero}\big\}. $$ for $ C\frac{\log(n)}{n} \le p \le \frac{1}{C}$ for some large constant $C>1$. In this paper, we setted this conjecture in the sparse regime when $p$ satisfies $$ 1 \le \liminf_{n\rightarrow \infty} \frac{pn}{\log(n)} \le \limsup_{n\rightarrow \infty} \frac{pn}{\log(n)} < + \infty. $$