77 research outputs found

### 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

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