On the Spielman-Teng Conjecture

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.