[{"citation":{"ista":"Erdös L, Xu Y. 2023. Small deviation estimates for the largest eigenvalue of Wigner matrices. Bernoulli. 29(2), 1063–1079.","apa":"Erdös, L., & Xu, Y. (2023). Small deviation estimates for the largest eigenvalue of Wigner matrices. Bernoulli. Bernoulli Society for Mathematical Statistics and Probability. https://doi.org/10.3150/22-BEJ1490","ieee":"L. Erdös and Y. Xu, “Small deviation estimates for the largest eigenvalue of Wigner matrices,” Bernoulli, vol. 29, no. 2. Bernoulli Society for Mathematical Statistics and Probability, pp. 1063–1079, 2023.","ama":"Erdös L, Xu Y. Small deviation estimates for the largest eigenvalue of Wigner matrices. Bernoulli. 2023;29(2):1063-1079. doi:10.3150/22-BEJ1490","chicago":"Erdös, László, and Yuanyuan Xu. “Small Deviation Estimates for the Largest Eigenvalue of Wigner Matrices.” Bernoulli. Bernoulli Society for Mathematical Statistics and Probability, 2023. https://doi.org/10.3150/22-BEJ1490.","mla":"Erdös, László, and Yuanyuan Xu. “Small Deviation Estimates for the Largest Eigenvalue of Wigner Matrices.” Bernoulli, vol. 29, no. 2, Bernoulli Society for Mathematical Statistics and Probability, 2023, pp. 1063–79, doi:10.3150/22-BEJ1490.","short":"L. Erdös, Y. Xu, Bernoulli 29 (2023) 1063–1079."},"publication":"Bernoulli","page":"1063-1079","article_type":"original","date_published":"2023-05-01T00:00:00Z","scopus_import":"1","article_processing_charge":"No","day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"12707","intvolume":" 29","title":"Small deviation estimates for the largest eigenvalue of Wigner matrices","status":"public","oa_version":"Preprint","type":"journal_article","issue":"2","abstract":[{"lang":"eng","text":"We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2112.12093"}],"oa":1,"external_id":{"isi":["000947270100008"],"arxiv":["2112.12093 "]},"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"quality_controlled":"1","isi":1,"doi":"10.3150/22-BEJ1490","language":[{"iso":"eng"}],"publication_identifier":{"issn":["1350-7265"]},"month":"05","year":"2023","department":[{"_id":"LaEr"}],"publisher":"Bernoulli Society for Mathematical Statistics and Probability","publication_status":"published","author":[{"first_name":"László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","full_name":"Erdös, László"},{"first_name":"Yuanyuan","last_name":"Xu","id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3","orcid":"0000-0003-1559-1205","full_name":"Xu, Yuanyuan"}],"volume":29,"date_created":"2023-03-05T23:01:05Z","date_updated":"2023-10-04T10:21:07Z","ec_funded":1},{"type":"journal_article","issue":"1","abstract":[{"lang":"eng","text":"We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"14775","intvolume":" 33","title":"Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices","status":"public","oa_version":"Preprint","scopus_import":"1","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"article_processing_charge":"No","day":"01","citation":{"ama":"Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 2023;33(1):677-725. doi:10.1214/22-aap1826","apa":"Schnelli, K., & Xu, Y. (2023). Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1826","ieee":"K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices,” The Annals of Applied Probability, vol. 33, no. 1. Institute of Mathematical Statistics, pp. 677–725, 2023.","ista":"Schnelli K, Xu Y. 2023. Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices. The Annals of Applied Probability. 33(1), 677–725.","short":"K. Schnelli, Y. Xu, The Annals of Applied Probability 33 (2023) 677–725.","mla":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied Probability, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp. 677–725, doi:10.1214/22-aap1826.","chicago":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1826."},"publication":"The Annals of Applied Probability","page":"677-725","article_type":"original","date_published":"2023-02-01T00:00:00Z","ec_funded":1,"acknowledgement":"K. Schnelli was supported by the Swedish Research Council Grants VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Y. Xu was supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond” No. 101020331.","year":"2023","publisher":"Institute of Mathematical Statistics","department":[{"_id":"LaEr"}],"publication_status":"published","author":[{"orcid":"0000-0003-0954-3231","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","first_name":"Kevin","full_name":"Schnelli, Kevin"},{"first_name":"Yuanyuan","last_name":"Xu","id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3","orcid":"0000-0003-1559-1205","full_name":"Xu, Yuanyuan"}],"volume":33,"date_updated":"2024-01-10T13:31:46Z","date_created":"2024-01-10T09:23:31Z","publication_identifier":{"issn":["1050-5164"]},"month":"02","external_id":{"isi":["000946432400021"],"arxiv":["2108.02728"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2108.02728"}],"project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331"}],"quality_controlled":"1","isi":1,"doi":"10.1214/22-aap1826","language":[{"iso":"eng"}]},{"scopus_import":"1","has_accepted_license":"1","article_processing_charge":"No","day":"01","page":"839-907","article_type":"original","citation":{"apa":"Schnelli, K., & Xu, Y. (2022). Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-022-04377-y","ieee":"K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices,” Communications in Mathematical Physics, vol. 393. Springer Nature, pp. 839–907, 2022.","ista":"Schnelli K, Xu Y. 2022. Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices. Communications in Mathematical Physics. 393, 839–907.","ama":"Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices. Communications in Mathematical Physics. 2022;393:839-907. doi:10.1007/s00220-022-04377-y","chicago":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Wigner Matrices.” Communications in Mathematical Physics. Springer Nature, 2022. https://doi.org/10.1007/s00220-022-04377-y.","short":"K. Schnelli, Y. Xu, Communications in Mathematical Physics 393 (2022) 839–907.","mla":"Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Wigner Matrices.” Communications in Mathematical Physics, vol. 393, Springer Nature, 2022, pp. 839–907, doi:10.1007/s00220-022-04377-y."},"publication":"Communications in Mathematical Physics","date_published":"2022-07-01T00:00:00Z","type":"journal_article","abstract":[{"lang":"eng","text":"We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+\\omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+\\omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles."}],"intvolume":" 393","ddc":["510"],"title":"Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner matrices","status":"public","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"11332","oa_version":"Published Version","file":[{"date_updated":"2022-08-05T06:01:13Z","date_created":"2022-08-05T06:01:13Z","success":1,"checksum":"bee0278c5efa9a33d9a2dc8d354a6c51","file_id":"11726","relation":"main_file","creator":"dernst","content_type":"application/pdf","file_size":1141462,"file_name":"2022_CommunMathPhys_Schnelli.pdf","access_level":"open_access"}],"publication_identifier":{"eissn":["1432-0916"],"issn":["0010-3616"]},"month":"07","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"isi":1,"quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"external_id":{"arxiv":["2102.04330"],"isi":["000782737200001"]},"language":[{"iso":"eng"}],"doi":"10.1007/s00220-022-04377-y","ec_funded":1,"file_date_updated":"2022-08-05T06:01:13Z","publisher":"Springer Nature","department":[{"_id":"LaEr"}],"publication_status":"published","year":"2022","acknowledgement":"Kevin Schnelli is supported in parts by the Swedish Research Council Grant VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Yuanyuan Xu is supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond” No. 101020331.","volume":393,"date_updated":"2023-08-03T06:34:24Z","date_created":"2022-04-24T22:01:44Z","author":[{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0954-3231","first_name":"Kevin","last_name":"Schnelli","full_name":"Schnelli, Kevin"},{"id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3","last_name":"Xu","first_name":"Yuanyuan","full_name":"Xu, Yuanyuan"}]},{"publication":"Journal of Mathematical Physics","citation":{"apa":"Cipolloni, G., Erdös, L., Schröder, D. J., & Xu, Y. (2022). Directional extremal statistics for Ginibre eigenvalues. Journal of Mathematical Physics. AIP Publishing. https://doi.org/10.1063/5.0104290","ieee":"G. Cipolloni, L. Erdös, D. J. Schröder, and Y. Xu, “Directional extremal statistics for Ginibre eigenvalues,” Journal of Mathematical Physics, vol. 63, no. 10. AIP Publishing, 2022.","ista":"Cipolloni G, Erdös L, Schröder DJ, Xu Y. 2022. Directional extremal statistics for Ginibre eigenvalues. Journal of Mathematical Physics. 63(10), 103303.","ama":"Cipolloni G, Erdös L, Schröder DJ, Xu Y. Directional extremal statistics for Ginibre eigenvalues. Journal of Mathematical Physics. 2022;63(10). doi:10.1063/5.0104290","chicago":"Cipolloni, Giorgio, László Erdös, Dominik J Schröder, and Yuanyuan Xu. “Directional Extremal Statistics for Ginibre Eigenvalues.” Journal of Mathematical Physics. AIP Publishing, 2022. https://doi.org/10.1063/5.0104290.","short":"G. Cipolloni, L. Erdös, D.J. Schröder, Y. Xu, Journal of Mathematical Physics 63 (2022).","mla":"Cipolloni, Giorgio, et al. “Directional Extremal Statistics for Ginibre Eigenvalues.” Journal of Mathematical Physics, vol. 63, no. 10, 103303, AIP Publishing, 2022, doi:10.1063/5.0104290."},"article_type":"original","date_published":"2022-10-14T00:00:00Z","scopus_import":"1","keyword":["Mathematical Physics","Statistical and Nonlinear Physics"],"day":"14","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","_id":"12243","title":"Directional extremal statistics for Ginibre eigenvalues","status":"public","ddc":["510","530"],"intvolume":" 63","file":[{"creator":"dernst","file_size":7356807,"content_type":"application/pdf","file_name":"2022_JourMathPhysics_Cipolloni2.pdf","access_level":"open_access","date_updated":"2023-01-30T08:01:10Z","date_created":"2023-01-30T08:01:10Z","success":1,"checksum":"2db278ae5b07f345a7e3fec1f92b5c33","file_id":"12436","relation":"main_file"}],"oa_version":"Published Version","type":"journal_article","abstract":[{"lang":"eng","text":"We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. "}],"issue":"10","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"external_id":{"isi":["000869715800001"],"arxiv":["2206.04443"]},"isi":1,"quality_controlled":"1","project":[{"call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d"}],"doi":"10.1063/5.0104290","language":[{"iso":"eng"}],"month":"10","publication_identifier":{"issn":["0022-2488"],"eissn":["1089-7658"]},"acknowledgement":"The authors are grateful to G. Akemann for bringing Refs. 19 and 24–26 to their attention. Discussions with Guillaume Dubach on a preliminary version of this project are acknowledged.\r\nL.E. and Y.X. were supported by the ERC Advanced Grant “RMTBeyond” under Grant No. 101020331. D.S. was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.","year":"2022","publication_status":"published","department":[{"_id":"LaEr"}],"publisher":"AIP Publishing","author":[{"orcid":"0000-0002-4901-7992","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","last_name":"Cipolloni","first_name":"Giorgio","full_name":"Cipolloni, Giorgio"},{"orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","first_name":"László","full_name":"Erdös, László"},{"last_name":"Schröder","first_name":"Dominik J","orcid":"0000-0002-2904-1856","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"},{"id":"7902bdb1-a2a4-11eb-a164-c9216f71aea3","first_name":"Yuanyuan","last_name":"Xu","full_name":"Xu, Yuanyuan"}],"date_updated":"2023-08-04T09:40:02Z","date_created":"2023-01-16T09:52:58Z","volume":63,"article_number":"103303","file_date_updated":"2023-01-30T08:01:10Z","ec_funded":1}]