Local Laws and Edge Universality for Noncentral Sample Covariance Matrices
Abstract
We consider the real noncentral sample covariance matrices 𝒲=YY⊤ with Y=A+Σ1/2X. Here A∈ℝM× N is deterministic, Σ∈ ℝM× M is a deterministic positive definite population covariance matrix and X∈ℝM× N has independent centered entries with variance N−1. We prove local laws near regular right edges down to optimal spectral scales without requiring the commutativity of AA⊤ and Σ. As a consequence, we obtain optimal eigenvalue rigidity at the rightmost regular edge and delocalization of the corresponding left and right singular vectors. We also show that, with high probability, there are no eigenvalues in the adjacent spectral gap beyond the optimal N−2/3 edge scale, up to an arbitrarily small Nε loss. Finally, we establish edge universality at the rightmost regular edge: after centering and scaling, the largest eigenvalue converges to the Tracy--Widom distribution. The main technical ingredient is a stability analysis of the matrix Dyson equation (MDE) associated with the linearization of Y, whose self-energy operator does not satisfy the flatness condition of the general MDE theory. Exploiting the special block structure, we reduce the stability analysis exactly to a two-dimensional operator. This reduction yields regularity of the spectral density and square-root behavior at regular right edges, together with sharp stability bounds near such edges.
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BibTeX
@misc{hu2026local,
title = {{Local Laws and Edge Universality for Noncentral Sample Covariance Matrices}},
author = {Can Hu and Jiang Hu and Zhidong Bai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27307},
eprint = {2608.27307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27307},
}
RIS
TY - UNPB
TI - Local Laws and Edge Universality for Noncentral Sample Covariance Matrices
AU - Can Hu
AU - Jiang Hu
AU - Zhidong Bai
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2608.27307
UR - https://arxiv.org/abs/2608.27307
AB - We consider the real noncentral sample covariance matrices 𝒲=YY^⊤ with Y=A+Σ^1/2X. Here A∈ℝ^M× N is deterministic, Σ∈ ℝ^M× M is a deterministic positive definite population covariance matrix and X∈ℝ^M× N has independent centered entries with variance N^−1. We prove local laws near regular right edges down to optimal spectral scales without requiring the commutativity of AA^⊤ and Σ. As a consequence, we obtain optimal eigenvalue rigidity at the rightmost regular edge and delocalization of the corresponding left and right singular vectors. We also show that, with high probability, there are no eigenvalues in the adjacent spectral gap beyond the optimal N^−2/3 edge scale, up to an arbitrarily small N^ε loss. Finally, we establish edge universality at the rightmost regular edge: after centering and scaling, the largest eigenvalue converges to the Tracy--Widom distribution. The main technical ingredient is a stability analysis of the matrix Dyson equation (MDE) associated with the linearization of Y, whose self-energy operator does not satisfy the flatness condition of the general MDE theory. Exploiting the special block structure, we reduce the stability analysis exactly to a two-dimensional operator. This reduction yields regularity of the spectral density and square-root behavior at regular right edges, together with sharp stability bounds near such edges.
ER -