Abstract
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.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Cited by
19 articles.
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