Least singular values of Gaussian matrices with deterministic shifts

Abstract

Let $A$ be a deterministic real square matrix and $G$ a standard Gaussian matrix. We determine $\E\smin(A+G)$, up to universal constants, in terms of the singular values of $A$. The expectation is comparable to the square root of the smallest zero of the expected characteristic polynomial of $(A+G{)}^{\top }(A+G)$. This resolves Problem~3 in Tikhomirov's ICM 2022 survey. We also prove that the zeros of the expected characteristic polynomial are majorized by the expected ordered eigenvalues of $(A+G{)}^{\top }(A+G)$.