The least singular value of a random symmetric matrix

Abstract

Let $A$ be a $n\times n$ symmetric matrix with $(A_{i,j}{)}_{i\le j}$, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}({\sigma }_{\min }(A)\le \epsilon /\sqrt{n})\le C\epsilon +e^{-cn},$$ where ${\sigma }_{\min }(A)$ denotes the least singular value of $A$ and the constants $C,c>0$ depend only on the distribution of the entries of $A$. This result confirms a folklore conjecture on the lower-tail asymptotics of the least singular value of random symmetric matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability $A$ has a repeated eigenvalue is $e^{-\Omega (n)}$, thus confirming a conjecture of Nguyen, Tao and Vu.

39 pages + 33 page Appendix. The appendix builds on our previous work arXiv:2105.11384 and provides the proof of a technical quasi-randomness statement. Accepted to Forum of Mathematics, Pi