{"title":"Functional central limit theorems for Wigner matrices","quality_controlled":"1","status":"public","oa_version":"Preprint","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-03-26T22:01:08Z","publisher":"Institute of Mathematical Statistics","scopus_import":"1","publication_status":"published","date_published":"2023-02-01T00:00:00Z","issue":"1","department":[{"_id":"LaEr"}],"abstract":[{"lang":"eng","text":"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 tracial mode, Trf (W), which is equivalent to the customary linear statistics of eigenvalues, we show that Trf (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\r\nthe 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 averaging. 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\r\nmultiresolvent local laws with traceless deterministic matrices from the companion paper (Comm. Math. Phys. 388 (2021) 1005–1048)."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2012.13218"}],"type":"journal_article","day":"01","month":"02","article_processing_charge":"No","date_updated":"2023-10-17T12:48:52Z","isi":1,"acknowledgement":"The second author is partially funded by the ERC Advanced Grant “RMTBEYOND” No. 101020331. The third author is supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation.","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020"}],"intvolume":"        33","volume":33,"year":"2023","citation":{"ieee":"G. Cipolloni, L. Erdös, and D. J. Schröder, “Functional central limit theorems for Wigner matrices,” <i>Annals of Applied Probability</i>, vol. 33, no. 1. Institute of Mathematical Statistics, pp. 447–489, 2023.","mla":"Cipolloni, Giorgio, et al. “Functional Central Limit Theorems for Wigner Matrices.” <i>Annals of Applied Probability</i>, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp. 447–89, doi:<a href=\"https://doi.org/10.1214/22-AAP1820\">10.1214/22-AAP1820</a>.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Annals of Applied Probability 33 (2023) 447–489.","ama":"Cipolloni G, Erdös L, Schröder DJ. Functional central limit theorems for Wigner matrices. <i>Annals of Applied Probability</i>. 2023;33(1):447-489. doi:<a href=\"https://doi.org/10.1214/22-AAP1820\">10.1214/22-AAP1820</a>","chicago":"Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Functional Central Limit Theorems for Wigner Matrices.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2023. <a href=\"https://doi.org/10.1214/22-AAP1820\">https://doi.org/10.1214/22-AAP1820</a>.","ista":"Cipolloni G, Erdös L, Schröder DJ. 2023. Functional central limit theorems for Wigner matrices. Annals of Applied Probability. 33(1), 447–489.","apa":"Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2023). Functional central limit theorems for Wigner matrices. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/22-AAP1820\">https://doi.org/10.1214/22-AAP1820</a>"},"article_type":"original","ec_funded":1,"_id":"12761","language":[{"iso":"eng"}],"author":[{"first_name":"Giorgio","last_name":"Cipolloni","full_name":"Cipolloni, Giorgio","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4901-7992"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","first_name":"László"},{"orcid":"0000-0002-2904-1856","full_name":"Schröder, Dominik J","id":"408ED176-F248-11E8-B48F-1D18A9856A87","last_name":"Schröder","first_name":"Dominik J"}],"publication":"Annals of Applied Probability","doi":"10.1214/22-AAP1820","page":"447-489","external_id":{"isi":["000946432400015"],"arxiv":["2012.13218"]},"publication_identifier":{"issn":["1050-5164"]}}