--- _id: '2225' abstract: - lang: eng 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" article_number: '33' author: - first_name: Alex full_name: Bloemendal, Alex last_name: Bloemendal - first_name: László full_name: Erdös, László id: 4DBD5372-F248-11E8-B48F-1D18A9856A87 last_name: Erdös orcid: 0000-0001-5366-9603 - first_name: Antti full_name: Knowles, Antti last_name: Knowles - first_name: Horng full_name: Yau, Horng last_name: Yau - first_name: Jun full_name: Yin, Jun last_name: Yin citation: ama: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 2014;19. doi:10.1214/EJP.v19-3054 apa: 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. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v19-3054 chicago: Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2014. https://doi.org/10.1214/EJP.v19-3054. ieee: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local laws for sample covariance and generalized Wigner matrices,” Electronic Journal of Probability, vol. 19. Institute of Mathematical Statistics, 2014. 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. mla: Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability, vol. 19, 33, Institute of Mathematical Statistics, 2014, doi:10.1214/EJP.v19-3054. short: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014). date_created: 2018-12-11T11:56:25Z date_published: 2014-03-15T00:00:00Z date_updated: 2021-01-12T06:56:07Z day: '15' ddc: - '510' department: - _id: LaEr doi: 10.1214/EJP.v19-3054 ec_funded: 1 file: - access_level: open_access checksum: 7eb297ff367a2ee73b21b6dd1e1948e4 content_type: application/pdf creator: system date_created: 2018-12-12T10:14:06Z date_updated: 2020-07-14T12:45:34Z file_id: '5055' file_name: IST-2016-427-v1+1_3054-16624-4-PB.pdf file_size: 810150 relation: main_file file_date_updated: 2020-07-14T12:45:34Z has_accepted_license: '1' intvolume: ' 19' language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '03' oa: 1 oa_version: Published Version project: - _id: 258DCDE6-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '338804' name: Random matrices, universality and disordered quantum systems publication: Electronic Journal of Probability publication_identifier: issn: - '10836489' publication_status: published publisher: Institute of Mathematical Statistics publist_id: '4739' pubrep_id: '427' quality_controlled: '1' status: public title: Isotropic local laws for sample covariance and generalized Wigner matrices tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 19 year: '2014' ...