Simplicity of singular value spectrum of random matrices and two-point quantitative invertibility

Let $A$ be an $n\times n$ random matrix with independent, identically distributed mean 0, variance 1 subgaussian entries. We prove that $$ \mathbb{P}(A\text{ has distinct singular values})\geq 1-e^{-cn} $$ for some $c>0$, confirming a conjecture of Vu. This result is then generalized to singular values of rectangular random matrices with i.i.d. entries. We also prove that for two fixed real numbers $\lambda_1,\lambda_2$ with a sufficient lower bound on $|\lambda_1-\lambda_2|$, we have a joint singular value small ball estimate for any $\epsilon>0$ $$ \mathbb{P}(\sigma_{min}(A-\lambda_1I_n)\leq\epsilon n^{-1/2},\sigma_{min}(A-\lambda_2I_n)\leq\epsilon n^{-1/2})\leq C\epsilon^2+e^{-cn}, $$ where $\sigma_{min}(A)$ is the minimal singular value of a square matrix $A$ and $I_n$ is the identity matrix. For much smaller $|\lambda_1-\lambda_2|$ we derive a similar estimate with $C$ replaced by $C\sqrt{n}/|\lambda_1-\lambda_2|$. This generalizes the one-point estimate of Rudelson and Vershynin, which proves $\mathbb{P}(\sigma_{min}(A)\leq \epsilon n^{-1/2})\leq C\epsilon+e^{-cn}$. Analogous two-point bounds are proven when $A$ has i.i.d. real and complex parts, with $\epsilon^4$ in place of $\epsilon^2$ on the right hand side of the estimate and for any complex numbers $\lambda_1,\lambda_2$. These two point estimates can be used to derive strong anticoncentration bounds for an arbitrary linear combination of two eigenvalues of $A$.