{"abstract":[{"lang":"eng","text":"We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij|2. As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W ≫N1-εn with some εn &gt; 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17, 19, 6]."}],"quality_controlled":"1","scopus_import":1,"department":[{"_id":"LaEr"}],"citation":{"ista":"Erdös L, Knowles A, Yau H, Yin J. 2013. The local semicircle law for a general class of random matrices. Electronic Journal of Probability. 18(59), 1–58.","short":"L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 18 (2013) 1–58.","ama":"Erdös L, Knowles A, Yau H, Yin J. The local semicircle law for a general class of random matrices. <i>Electronic Journal of Probability</i>. 2013;18(59):1-58. doi:<a href=\"https://doi.org/10.1214/EJP.v18-2473\">10.1214/EJP.v18-2473</a>","chicago":"Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “The Local Semicircle Law for a General Class of Random Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2013. <a href=\"https://doi.org/10.1214/EJP.v18-2473\">https://doi.org/10.1214/EJP.v18-2473</a>.","ieee":"L. Erdös, A. Knowles, H. Yau, and J. Yin, “The local semicircle law for a general class of random matrices,” <i>Electronic Journal of Probability</i>, vol. 18, no. 59. Institute of Mathematical Statistics, pp. 1–58, 2013.","apa":"Erdös, L., Knowles, A., Yau, H., &#38; Yin, J. (2013). The local semicircle law for a general class of random matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/EJP.v18-2473\">https://doi.org/10.1214/EJP.v18-2473</a>","mla":"Erdös, László, et al. “The Local Semicircle Law for a General Class of Random Matrices.” <i>Electronic Journal of Probability</i>, vol. 18, no. 59, Institute of Mathematical Statistics, 2013, pp. 1–58, doi:<a href=\"https://doi.org/10.1214/EJP.v18-2473\">10.1214/EJP.v18-2473</a>."},"intvolume":"        18","author":[{"last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","first_name":"László"},{"full_name":"Knowles, Antti","last_name":"Knowles","first_name":"Antti"},{"first_name":"Horng","full_name":"Yau, Horng","last_name":"Yau"},{"full_name":"Yin, Jun","last_name":"Yin","first_name":"Jun"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"ddc":["530"],"title":"The local semicircle law for a general class of random matrices","status":"public","has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2013-05-29T00:00:00Z","file_date_updated":"2020-07-14T12:45:50Z","oa_version":"Published Version","date_created":"2018-12-11T11:59:51Z","month":"05","type":"journal_article","_id":"2837","publication":"Electronic Journal of Probability","date_updated":"2021-01-12T07:00:06Z","file":[{"checksum":"aac9e52a00cb2f5149dc9e362b5ccf44","date_updated":"2020-07-14T12:45:50Z","access_level":"open_access","file_name":"IST-2016-406-v1+1_2473-13759-1-PB.pdf","creator":"system","relation":"main_file","file_id":"5169","date_created":"2018-12-12T10:15:46Z","file_size":651497,"content_type":"application/pdf"}],"page":"1-58","publist_id":"3962","doi":"10.1214/EJP.v18-2473","issue":"59","pubrep_id":"406","publication_status":"published","oa":1,"language":[{"iso":"eng"}],"day":"29","volume":18,"publisher":"Institute of Mathematical Statistics","year":"2013"}