The least singular value of a random symmetric matrix

TL;DR

This paper establishes sharp bounds on the probability that the smallest singular value of a random symmetric matrix is very small, confirming a longstanding conjecture and showing that repeated eigenvalues are exponentially rare.

Contribution

It proves the optimal lower-tail bound for the least singular value of symmetric matrices with subgaussian entries, resolving a folklore conjecture and confirming the rarity of repeated eigenvalues.

Findings

Probability of small least singular value is bounded by Cε + e^{-cn}

Repeated eigenvalues occur with exponentially small probability

Results are optimal up to distribution-dependent constants

Abstract

Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(\sigma_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C \varepsilon + e^{-cn},$$ 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.