Intermediate Singular Values of Random Matrices under Second-Moment and Anti-Concentration Assumptions
Abstract
Let A=(ξij) be an n× n random matrix with independent, not necessarily identically distributed, real entries satisfying 𝔼ξij=0, 𝔼ξij2=1, supz∈ℝℙ(|ξij−z|<a)≤ b for fixed a>0 and b∈(0,1). We prove that, for every δ∈(0,1), there are constants c,C>0, depending only on a,b,δ, such that ℙ( sn+1−l(A)>Ctl/√n ) ≤ exp (−cmin{tl,n}) for every t≥1 and every 1≤ l≤(1−δ)n. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order l/√n with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives sn+1−l(A)asymp l/√n with failure probability exponentially small in l. The same argument gives the rectangular scale √(N+1)−√(n−l+1) for N× n matrices whenever N−n+l≤(1−δ)N.
Related
- Least singular values of Gaussian matrices with deterministic shifts
- On the monotonicity of average singular values of complex Gaussian random matrices
- Meromorphic order from Taylor coefficients, least singular values of Hankel matrices and rational approximants
- A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices
- On a class of meromorphic functions with three singular values
BibTeX
@misc{polavarapu2026intermediate,
title = {{Intermediate Singular Values of Random Matrices under Second-Moment and Anti-Concentration Assumptions}},
author = {Achintya Raya Polavarapu and Manuel Fernandez V},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30722},
eprint = {2608.30722},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30722},
}
RIS
TY - UNPB
TI - Intermediate Singular Values of Random Matrices under Second-Moment and Anti-Concentration Assumptions
AU - Achintya Raya Polavarapu
AU - Manuel Fernandez V
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30722
UR - https://arxiv.org/abs/2608.30722
AB - Let A=(ξ_ij) be an n× n random matrix with independent, not necessarily identically distributed, real entries satisfying 𝔼ξ_ij=0, 𝔼ξ_ij^2=1, sup_z∈ℝℙ(|ξ_ij−z|<a)≤ b for fixed a>0 and b∈(0,1). We prove that, for every δ∈(0,1), there are constants c,C>0, depending only on a,b,δ, such that ℙ( s_n+1−l(A)>Ctl/√n ) ≤ exp (−cmin{tl,n}) for every t≥1 and every 1≤ l≤(1−δ)n. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order l/√n with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives s_n+1−l(A)asymp l/√n with failure probability exponentially small in l. The same argument gives the rectangular scale √(N+1)−√(n−l+1) for N× n matrices whenever N−n+l≤(1−δ)N.
ER -