The least singular value of a random symmetric matrix

Let A be an $n \times n$ symmetric matrix with $(A_{i,j})_{i\leqslant j}$ independent and identically distributed according to a subgaussian distribution. We show that $$ \begin{align*}\mathbb{P}(\sigma_{\min}(A) \leqslant \varepsilon n^{-1/2} ) \leqslant C \varepsilon + e^{-cn},\end{align*} $$ 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 the folklore conjecture on the lower tail of the least singular value of such 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 that A has a repeated eigenvalue is $e^{-\Omega (n)}$ , thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777–816].