{"project":[{"grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"volume":19,"type":"journal_article","citation":{"chicago":"Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2014. <a href=\"https://doi.org/10.1214/EJP.v19-3054\">https://doi.org/10.1214/EJP.v19-3054</a>.","short":"A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014).","apa":"Bloemendal, A., Erdös, L., Knowles, A., Yau, H., &#38; Yin, J. (2014). Isotropic local laws for sample covariance and generalized Wigner matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/EJP.v19-3054\">https://doi.org/10.1214/EJP.v19-3054</a>","ista":"Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. 2014. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 19, 33.","ama":"Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample covariance and generalized Wigner matrices. <i>Electronic Journal of Probability</i>. 2014;19. doi:<a href=\"https://doi.org/10.1214/EJP.v19-3054\">10.1214/EJP.v19-3054</a>","ieee":"A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local laws for sample covariance and generalized Wigner matrices,” <i>Electronic Journal of Probability</i>, vol. 19. Institute of Mathematical Statistics, 2014.","mla":"Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” <i>Electronic Journal of Probability</i>, vol. 19, 33, Institute of Mathematical Statistics, 2014, doi:<a href=\"https://doi.org/10.1214/EJP.v19-3054\">10.1214/EJP.v19-3054</a>."},"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","image":"/images/cc_by.png","short":"CC BY (4.0)"},"_id":"2225","date_published":"2014-03-15T00:00:00Z","status":"public","oa_version":"Published Version","pubrep_id":"427","publisher":"Institute of Mathematical Statistics","day":"15","intvolume":"        19","language":[{"iso":"eng"}],"year":"2014","quality_controlled":"1","publication_status":"published","file":[{"file_size":810150,"file_name":"IST-2016-427-v1+1_3054-16624-4-PB.pdf","file_id":"5055","relation":"main_file","date_updated":"2020-07-14T12:45:34Z","creator":"system","checksum":"7eb297ff367a2ee73b21b6dd1e1948e4","access_level":"open_access","date_created":"2018-12-12T10:14:06Z","content_type":"application/pdf"}],"file_date_updated":"2020-07-14T12:45:34Z","has_accepted_license":"1","publication":"Electronic Journal of Probability","department":[{"_id":"LaEr"}],"author":[{"last_name":"Bloemendal","full_name":"Bloemendal, Alex","first_name":"Alex"},{"orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","full_name":"Erdös, László"},{"last_name":"Knowles","full_name":"Knowles, Antti","first_name":"Antti"},{"full_name":"Yau, Horng","last_name":"Yau","first_name":"Horng"},{"first_name":"Jun","full_name":"Yin, Jun","last_name":"Yin"}],"publist_id":"4739","doi":"10.1214/EJP.v19-3054","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["10836489"]},"ddc":["510"],"title":"Isotropic local laws for sample covariance and generalized Wigner matrices","oa":1,"date_updated":"2021-01-12T06:56:07Z","ec_funded":1,"abstract":[{"text":"We consider sample covariance matrices of the form X∗X, where X is an M×N matrix with independent random entries.  We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent (X∗X−z)−1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.\r\n","lang":"eng"}],"article_number":"33","date_created":"2018-12-11T11:56:25Z","month":"03"}