# Singularity of sparse Bernoulli matrices

@article{Litvak2020SingularityOS, title={Singularity of sparse Bernoulli matrices}, author={Alexander E. Litvak and Konstantin E. Tikhomirov}, journal={arXiv: Probability}, year={2020} }

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}&=(1+o_n(1)){\mathbb P}\big\{\mbox{$M_n$ contains a zero row or column}\big\}\\ &=(2+o_n(1))n\,(1-p)^n, \end{align*} where $o_n(1)$ denotes a quantity which converges to zero as $n\to\infty$. We provide the corresponding upper and lower…

## 6 Citations

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

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This paper setted the conjecture that an A_n is singular matrix with i.i.d Bernoulli entries satisfies the sparse regime when p satisfies 1-o_n(1) for some large constant $C>1$.

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