Ultra-hard-edge universality under deterministic preconditioning
Abstract
Let An be an n× n real matrix with iid entries distributed as a fixed centered, variance-one, symmetric subgaussian random variable. For deterministic invertible Mn, set hn=|Mn−1|HS and Bn=Mn−1/hn. We prove ℙ(smin(MnAn)≤ε/hn) =(1+o(1))εΘn(Bn) as n→∞, uniformly over Mn and e−cn≤ε≤ n−C. Here Θn(B)=𝔼√(nU⊤ BB⊤ U), where U is uniform on Sn−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.
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BibTeX
@misc{haggblom2026ultrahardedge,
title = {{Ultra-hard-edge universality under deterministic preconditioning}},
author = {Christian E. Häggblom},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31143},
eprint = {2610.31143},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31143},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ultra-hard-edge universality under deterministic preconditioning
AU - Christian E. Häggblom
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31143
UR - https://arxiv.org/abs/2610.31143
AB - Let A_n be an n× 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|_HS and B_n=M_n^−1/h_n. We prove ℙ(s_min(M_nA_n)≤ε/h_n) =(1+o(1))εΘ_n(B_n) as n→∞, uniformly over M_n and e^−cn≤ε≤ n^−C. Here Θ_n(B)=𝔼√(nU^⊤ BB^⊤ 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.
ER -