Rank deficiency of random matrices

Vishesh Jain; Ashwin Sah; Mehtaab Sawhney

doi:10.1214/22-ECP455

2022 Rank deficiency of random matrices

Vishesh Jain, Ashwin Sah, Mehtaab Sawhney

Author Affiliations +

Electron. Commun. Probab. 27: 1-9 (2022). DOI: 10.1214/22-ECP455

Abstract

Let ${M_{n}}$ be a random $n\times n$ matrix with i.i.d. $\operatorname{Bernoulli}(1/ 2)$ entries. We show that for fixed $k\ge 1$ ,

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}{\log _{2}}\mathbb{P}[\operatorname{corank}{M_{n}}\ge k]=-k.$

Acknowledgments

We thank Jordan Ellenberg for a Twitter question which motivated the present work. We also thank Han Huang for sharing a preliminary version of [6] with us. Finally we are grateful to the anonymous referees for their careful reading of our paper and for numerous comments which have improved the presentation.

Citation

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Vishesh Jain. Ashwin Sah. Mehtaab Sawhney. "Rank deficiency of random matrices." Electron. Commun. Probab. 27 1 - 9, 2022. https://doi.org/10.1214/22-ECP455