  Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$
and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$.
We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) =
(1+o(1)) \varepsilon + e^{-\Omega(n)}\] for all $0 \leq \varepsilon \ll 1$.
This resolves, up to a $1+o(1)$ factor, a seminal conjecture of Spielman and
Teng.
