Singularity of random symmetric matrices -- a combinatorial approach to improved bounds

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 $-1$ with probability $1/2$ each). It is widely conjectured that $M_n$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_n$, due to Vershynin (2011), is $2^{-n^c}$, for some unspecified small constant $c > 0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_n$ is at most $2^{-n^{1/4}\sqrt{\log{n}}/1000}$ for all sufficiently large $n$. 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.Comment: Final version incorporating referee comment

Quantitative invertibility of random matrices: a combinatorial perspective

We study the lower tail behavior of the least singular value of an $n\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix. This setting has previously been considered in a series of influential works by Tao and Vu, most notably in connection with the strong circular law, and the smoothed analysis of the condition number, and our results improve upon theirs in two ways: (i) We are able to handle $\|M\| = O(\exp(n^{c}))$, whereas the results of Tao and Vu are applicable only for $M = O(\text{poly(n)})$. (ii) Even for $M = O(\text{poly(n)})$, we are able to extract more refined information -- for instance, our results show that for such $M$, the probability that $M_n$ is singular is $O(\exp(-n^{c}))$, whereas even in the case when $\xi$ is a Bernoulli random variable, the results of Tao and Vu only give a bound of the form $O_{C}(n^{-C})$ for any constant $C>0$. As opposed to all previous works obtaining such bounds with error rate better than $n^{-1}$, our proof makes no use either of the inverse Littlewood--Offord theorems, or of any sophisticated net constructions. Instead, we show how to reduce the problem from the (complex) sphere to (Gaussian) integer vectors, where it is solved directly by utilizing and extending a combinatorial approach to the singularity problem for random discrete matrices, recently developed by Ferber, Luh, Samotij, and the author.Comment: 37 pages; comments welcome. arXiv admin note: text overlap with arXiv:1904.1110