450 research outputs found
Singularity of random symmetric matrices -- a combinatorial approach to improved bounds
Let denote a random symmetric matrix whose upper diagonal
entries are independent and identically distributed Bernoulli random variables
(which take values and with probability each). It is widely
conjectured that is singular with probability at most . On
the other hand, the best known upper bound on the singularity probability of
, due to Vershynin (2011), is , for some unspecified small
constant . 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 is at most for all sufficiently
large . 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
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