• Corpus ID: 221995957

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

@article{Huang2020SingularityOB,
title={Singularity of Bernoulli matrices in the sparse regime \$pn = O(\log(n))\$},
author={Han Huang},
journal={arXiv: Probability},
year={2020}
}
• Han Huang
• Published 29 September 2020
• Mathematics
• arXiv: Probability
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…
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