FUNCTIONAL CENTRAL LIMIT THEOREMS FOR WIGNER MATRICES

• 成果类型: Article
• 署名作者: Cipolloni, Giorgio; Erdos, Laszlo; Schroder, Dominik
• 署名单位: Institute of Science & Technology - Austria; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1820
• 发表日期: 2023
• 页码: 447-489
• 关键词: linear eigenvalue statistics; altshuler-shklovskii formulas; spectral statistics; general beta; fluctuations; clt; UNIVERSALITY; CONVERGENCE; RIGIDITY

摘要

We consider the fluctuations of regular functions f of a Wigner matrix W viewed as an entire matrix f (W). Going beyond the well-studied tra-cial mode, Tr f (W), which is equivalent to the customary linear statistics of eigenvalues, we show that Tr f (W)A is asymptotically normal for any nontrivial bounded deterministic matrix A. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of f (W) in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. As a main motivation to study CLT in such generality on small mesoscopic scales, we determine the fluctuations in the eigenstate thermalization hypothesis (Phys. Rev. A 43 (1991) 2046-2049), that is, prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral av-eraging. Finally, in the macroscopic regime our result also generalizes (Zh. Mat. Fiz. Anal. Geom. 9 (2013) 536-581, 611, 615) to complex W and to all crossover ensembles in between. The main technical inputs are the recent multiresolvent local laws with traceless deterministic matrices from the com-panion paper