# 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} }

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

- 2020

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

- MathematicsProbability 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…

Singularity of random Bernoulli matrices

- Mathematics
- 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…

Structure of eigenvectors of random regular digraphs

- MathematicsTransactions of the American Mathematical Society
- 2019

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…

Circular law for sparse random regular digraphs

- Mathematics
- 2018

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.…

Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

- Mathematics
- 2018

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…

On the singularity probability of random Bernoulli matrices

- Mathematics
- 2005

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…

The rank of random regular digraphs of constant degree

- Mathematics, Computer ScienceJ. Complex.
- 2018

It is shown that A_n has rank at least at least $n-1$ with probability going to one as $n$ goes to infinity.

On the singularity probability of discrete random matrices

- Mathematics
- 2009

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…