# Singularity of random Bernoulli matrices

@article{Tikhomirov2018SingularityOR,
title={Singularity of random Bernoulli matrices},
author={Konstantin E. Tikhomirov},
journal={arXiv: Probability},
year={2018}
}
For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that ${\mathbb P}\{\mbox{$M_n$is singular}\}=(1/2+o_n(1))^n$, which settles an old problem. Some generalizations are considered.
57 Citations
Singularity of random symmetric matrices revisited
• Mathematics, Computer Science
• 2020
In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $\exp(-c n^{1/2})$ on the singularity probability, the method is different and considerably simpler.
On the permanent of a random symmetric matrix
• Mathematics
Selecta Mathematica
• 2021
Let $M_{n}$ denote a random symmetric $n\times n$ matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values $\pm 1$ with probability $1/2$ each). Resolving
Singularity of sparse Bernoulli matrices
• Mathematics, Computer Science
• 2020
There is a universal constant C\geq 1 such that, whenever $p and$n$satisfy C\log n/n/n\leq p-1, there is a singular value of$M_n$such that it contains a zero row or column. Singularity of Bernoulli matrices in the sparse regime$pn = O(\log(n))$This paper setted the conjecture that an A_n is singular matrix with i.i.d Bernoulli entries satisfies the sparse regime when p satisfies 1-o_n(1) for some large constant$C>1$. SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO IMPROVED BOUNDS • Mathematics, Computer Science Forum of Mathematics, Sigma • 2019 The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij, and improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005). On the singularity of random symmetric matrices • Mathematics • 2019 A well-known conjecture states that a random symmetric$n \times n$matrix with entries in$\{-1,1\}$is singular with probability$\Theta\big( n^2 2^{-n} \big)$. In this paper we prove that the Singularity of discrete random matrices • Mathematics Geometric and Functional Analysis • 2021 Let$\xi$be a non-constant real-valued random variable with finite support, and let$M_{n}(\xi)$denote an$n\times n$random matrix with entries that are independent copies of$\xi$. For$\xi$Quantitative Group Testing and the rank of random matrices • Mathematics, Computer Science ArXiv • 2020 Using theoretical analysis and simulations, the modified algorithms solve the QGT problem for values of$ m $that are smaller than those required for the original algorithms. On the smallest singular value of symmetric random matrices • Mathematics, Computer Science Combinatorics, Probability and Computing • 2021 These methods suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices, and introduce new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold. Singularity of discrete random matrices II • Mathematics • 2020 Let$\xi$be a non-constant real-valued random variable with finite support, and let$M_{n}(\xi)$denote an$n\times n$random matrix with entries that are independent copies of$\xi\$. We show that,