SINGULARITY OF RANDOM SYMMETRIC MATRICES—A COMBINATORIAL APPROACH TO IMPROVED BOUNDS

Abstract

<jats:p>Let <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline1" xlink:type="simple" /><jats:tex-math>$M_{n}$</jats:tex-math></jats:alternatives> </jats:inline-formula> denote a random symmetric <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline2" xlink:type="simple" /><jats:tex-math>$n\times n$</jats:tex-math></jats:alternatives> </jats:inline-formula> matrix whose upper-diagonal entries are independent and identically distributed Bernoulli random variables (which take values <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline3" xlink:type="simple" /><jats:tex-math>$1$</jats:tex-math></jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline4" xlink:type="simple" /><jats:tex-math>$-1$</jats:tex-math></jats:alternatives> </jats:inline-formula> with probability <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline5" xlink:type="simple" /><jats:tex-math>$1/2$</jats:tex-math></jats:alternatives> </jats:inline-formula> each). It is widely conjectured that <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline6" xlink:type="simple" /><jats:tex-math>$M_{n}$</jats:tex-math></jats:alternatives> </jats:inline-formula> is singular with probability at most <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline7" xlink:type="simple" /><jats:tex-math>$(2+o(1))^{-n}$</jats:tex-math></jats:alternatives> </jats:inline-formula>. On the other hand, the best known upper bound on the singularity probability of <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline8" xlink:type="simple" /><jats:tex-math>$M_{n}$</jats:tex-math></jats:alternatives> </jats:inline-formula>, due to Vershynin (2011), is <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline9" xlink:type="simple" /><jats:tex-math>$2^{-n^{c}}$</jats:tex-math></jats:alternatives> </jats:inline-formula>, for some unspecified small constant <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline10" xlink:type="simple" /><jats:tex-math>$c&gt;0$</jats:tex-math></jats:alternatives> </jats:inline-formula>. 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 <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline11" xlink:type="simple" /><jats:tex-math>$M_{n}$</jats:tex-math></jats:alternatives> </jats:inline-formula> is at most <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline12" xlink:type="simple" /><jats:tex-math>$2^{-n^{1/4}\sqrt{\log n}/1000}$</jats:tex-math></jats:alternatives> </jats:inline-formula> for all sufficiently large <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S2050509419000215_inline13" xlink:type="simple" /><jats:tex-math>$n$</jats:tex-math></jats:alternatives> </jats:inline-formula>. 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.</jats:p>