Ultra-hard-edge universality under deterministic preconditioning

Abstract

Let $A_n$ be an $n\times n$ real matrix with iid entries distributed as a fixed centered, variance-one, symmetric subgaussian random variable. For deterministic invertible $M_n$, set $h_n=\|M_n^{-1}\|_{\mathrm{HS}}$ and $B_n=M_n^{-1}/h_n$. We prove $\mathbb{P}(s_{\min}(M_nA_n)\le\varepsilon/h_n) =(1+o(1))\varepsilonΘ_n(B_n)$ as $n\to\infty$, uniformly over $M_n$ and $e^{-cn}\le\varepsilon\le n^{-C}$. Here $Θ_n(B)=\mathbb{E}\sqrt{nU^\top BB^\top U}$, where $U$ is uniform on $S^{n-1}$. The constants $c>0$ and $C>1$ depend only on the entry law. No condition-number bound or effective-rank assumption is required. The coefficient tends to one for diffuse inverse profiles. For convergent concentrated or spiked profiles, its limit is an explicit Gaussian expectation. The proof combines the submicroscopic comparison of Sah, Sahasrabudhe, and Sawhney with signed-permutation averaging and a weighted comparison for random hyperplane normals. Applications include inverse amplification, square sample covariance matrices with arbitrary deterministic positive-definite population matrices, and independent random environments.