# Singularity of random symmetric matrices revisited

@article{Campos2020SingularityOR,
title={Singularity of random symmetric matrices revisited},
author={Marcelo Campos and Matthew Jenssen and Marcus Michelen and Julian Sahasrabudhe},
journal={arXiv: Probability},
year={2020}
}
Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n \times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp(-c(n\log n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. 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, our method is different and considerably simpler.
2 Citations
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#### References

SHOWING 1-10 OF 23 REFERENCES
Random symmetric matrices are almost surely nonsingular
• Mathematics
• 2005
Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ isExpand
Singularity of random Bernoulli matrices
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. SomeExpand
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 thatExpand
Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier resultExpand
SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO IMPROVED BOUNDS
• Mathematics
• Forum of Mathematics, Sigma
• 2019
Let $M_{n}$ denote a random symmetric $n\times n$ matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ withExpand
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 theExpand
On the smallest singular value of symmetric random matrices.
• Mathematics
• 2020
We show that for an $n\times n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean $0$ and variance $1$,Expand
On random ±1 matrices: Singularity and determinant
• Mathematics
• 2006
This papers contains two results concerning random n × n Bernoulli matrices. First, we show that with probability tending to 1 the determinant has absolute value $\sqrt{n!}\exp(O(\sqrt{n \ln n}))$.Expand
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 thisExpand
The Littlewood-Offord problem and invertibility of random matrices
• Mathematics
• 2007
Abstract We prove two basic conjectures on the distribution of the smallest singular value of random n × n matrices with independent entries. Under minimal moment assumptions, we show that theExpand