PreprintCC0arXiv
Tracy Widom law for sample covariance matrices under quadratic form concentration
Abstract
We study sample covariance matrices formed from i.i.d., centered, isotropic random vectors q1, …, qN ∈ ℝM, whose coordinates are allowed to be dependent. We assume the random vectors satisfy a concentration bound for their quadratic forms qiT A qi. Under this assumption, we establish edge universality and obtain the Tracy--Widom law for the largest eigenvalue of the sample covariance matrix after appropriate centering and scaling. In particular, our result removes the unconditionality assumption in existing edge universality results for log-concave ensembles~\cite{BX}, resolving the question that originally motivated our work. The concentration assumption also allows us to extend the result beyond the log-concave class.
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BibTeX
@misc{chen2026tracy,
title = {{Tracy Widom law for sample covariance matrices under quadratic form concentration}},
author = {Guangchen Chen and Xing Lyu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41416},
eprint = {2610.41416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41416},
note = {CC0},
}
RIS
TY - UNPB
TI - Tracy Widom law for sample covariance matrices under quadratic form concentration
AU - Guangchen Chen
AU - Xing Lyu
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41416
UR - https://arxiv.org/abs/2610.41416
AB - We study sample covariance matrices formed from i.i.d., centered, isotropic random vectors q_1, …, q_N ∈ ℝ^M, whose coordinates are allowed to be dependent. We assume the random vectors satisfy a concentration bound for their quadratic forms q_i^T A q_i. Under this assumption, we establish edge universality and obtain the Tracy--Widom law for the largest eigenvalue of the sample covariance matrix after appropriate centering and scaling. In particular, our result removes the unconditionality assumption in existing edge universality results for log-concave ensembles~\cite{BX}, resolving the question that originally motivated our work. The concentration assumption also allows us to extend the result beyond the log-concave class.
ER -