  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.
