{"day":"01","department":[{"_id":"LaEr"}],"abstract":[{"text":"Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).","lang":"eng"}],"page":"1270-1334","issue":"3","scopus_import":1,"month":"05","doi":"10.1214/18-AOP1284","publisher":"Institute of Mathematical Statistics","date_created":"2019-06-02T21:59:13Z","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","first_name":"Zhigang"},{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Schnelli","first_name":"Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","full_name":"Schnelli, Kevin","orcid":"0000-0003-0954-3231"}],"oa_version":"Preprint","project":[{"name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"quality_controlled":"1","language":[{"iso":"eng"}],"_id":"6511","external_id":{"arxiv":["1612.05920"]},"publication":"Annals of Probability","oa":1,"date_published":"2019-05-01T00:00:00Z","title":"Local single ring theorem on optimal scale","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-12T08:07:49Z","volume":47,"publication_identifier":{"issn":["00911798"]},"citation":{"mla":"Bao, Zhigang, et al. “Local Single Ring Theorem on Optimal Scale.” Annals of Probability, vol. 47, no. 3, Institute of Mathematical Statistics, 2019, pp. 1270–334, doi:10.1214/18-AOP1284.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Single Ring Theorem on Optimal Scale.” Annals of Probability. Institute of Mathematical Statistics, 2019. https://doi.org/10.1214/18-AOP1284.","short":"Z. Bao, L. Erdös, K. Schnelli, Annals of Probability 47 (2019) 1270–1334.","ama":"Bao Z, Erdös L, Schnelli K. Local single ring theorem on optimal scale. Annals of Probability. 2019;47(3):1270-1334. doi:10.1214/18-AOP1284","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2019). Local single ring theorem on optimal scale. Annals of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/18-AOP1284","ista":"Bao Z, Erdös L, Schnelli K. 2019. Local single ring theorem on optimal scale. Annals of Probability. 47(3), 1270–1334.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Local single ring theorem on optimal scale,” Annals of Probability, vol. 47, no. 3. Institute of Mathematical Statistics, pp. 1270–1334, 2019."},"publication_status":"published","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1612.05920"}],"year":"2019","type":"journal_article","article_processing_charge":"No","ec_funded":1,"intvolume":" 47"}