Phase transition for the smallest eigenvalue of covariance matrices

• 成果类型: Article
• 署名作者: Bao, Zhigang; Lee, Jaehun; Xu, Xiaocong
• 署名单位: Hong Kong University of Science & Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01298-w
• 发表日期: 2025
• 页码: 35-133
• 关键词: fixed-energy universality; tracy-widom distribution; Sufficient condition; linear statistics; edge universality; limit; fluctuations; CONVERGENCE; band; delocalization

摘要

In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices S(Y) = YY *, where Y = ( yi j) is an M x N matrix with iid mean 0 variance N -1 entries. We consider the regime M = M(N) and M/ N. c8. R\{1} as N. 8. It is known that for the extreme eigenvalues of Wigner matrices and the largest eigenvalue of S(Y), a weak 4thmoment condition is necessary and sufficient for the Tracy-Widom law (Ding and Yang in Ann Appl Probab 28(3):1679-1738, 2018. https:// doi.org/10.1214/ 17- AAP1341; Lee and Yin in Duke Math J 163(1):117-173, 2014. https:// doi.org/10.1215/ 00127094-2414767). In this paper, we show that the Tracy-Widom lawismore robust for the smallest eigenvalue of S(Y), by discovering a phase transition induced by the fatness of the tail of yi j 's. More specifically, we assume that yi j is symmetrically distributed with tail probability P(|v Nyi j| = x) x - a when x. 8, for some a. (2, 4). We show the following conclusions: (1) When a > 83, the smallest eigenvalue follows the Tracy-Widom law on scale N - 2 3; (2) When 2 < a < 83, the smallest eigenvalue follows the Gaussian law on scale N - a4; (3) When a = 83, the distribution is given by an interpolation between Tracy-Widom and Gaussian; (4) In case a = 10 3, in addition to the left edge of the MP law, a deterministic shift of order N1- a2 shall be subtracted from the smallest eigenvalue, in both the Tracy-Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by Aggarwal et al. (J Eur Math Soc 23(11):3707-3800, 2021. https:// doi. org/10.4171/ jems/1089) which is originally done for the bulk regime of the Levy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.