• 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 $$1 \le \liminf_{n\rightarrow \infty} \frac{pn}{\log(n… 3 Citations Singularity of discrete random matrices II • Mathematics • 2020 Let \xi be a non-constant real-valued random variable with finite support, and let M_{n}(\xi) denote an n\times n random matrix with entries that are independent copies of \xi. We show that, PR ] 1 3 O ct 2 02 0 SINGULARITY OF DISCRETE RANDOM MATRICES II Let ξ be a non-constant real-valued random variable with finite support, and let Mn(ξ) denote an n× n random matrix with entries that are independent copies of ξ. We show that, if ξ is not uniform on Rank deficiency of random matrices • Mathematics • 2021 Let Mn be a random n×n matrix with i.i.d. Bernoulli(1/2) entries. We show that for fixed k ≥ 1, lim n→∞ 1 n log 2 P[corankMn ≥ k] = −k. ## References SHOWING 1-10 OF 20 REFERENCES Singularity of sparse Bernoulli matrices • Mathematics • 2020 Let M_n be an n\times n random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant C\geq 1 such that, whenever p and n satisfy C\log n/n\leq p\leq C^{-1}, Adjacency matrices of random digraphs: singularity and anti-concentration • Mathematics • 2015 Let {\mathcal D}_{n,d} be the set of all d-regular directed graphs on n vertices. Let G be a graph chosen uniformly at random from {\mathcal D}_{n,d} and M be its adjacency matrix. We The smallest singular value of a shifted d-regular random square matrix • Mathematics Probability Theory and Related Fields • 2018 We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let$$C_1<d< c n/\log ^2
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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
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Let $n$ be a large integer, let $d$ satisfy $C\leq d\leq \exp(c\sqrt{\ln n})$ for some universal constants $c, C>0$, and let $z\in {\mathcal C}$. Further, denote by $M$ the adjacency matrix of a
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Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices.
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We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we
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Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that
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It is shown that A_n has rank at least at least $n-1$ with probability going to one as $n$ goes to infinity.
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Abstract Let n be a large integer and M n be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this