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LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE
Hong Kong Univ Sci & Technol, Dept Math, Kowloon, Clear Water Bay, Hong Kong, Peoples R China..
IST Austria, Campus 1, A-3400 Klosterneuburg, Austria..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2019 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 47, no 3, p. 1270-1334Article in journal (Refereed) Published
Abstract [en]

Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Sigma 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 : = U Sigma V* 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+epsilon and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).

Place, publisher, year, edition, pages
Institute of Mathematical Statistics, 2019. Vol. 47, no 3, p. 1270-1334
Keywords [en]
Non-Hermitian random matrices, local eigenvalue density, single ring theorem, free convolution
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-251704DOI: 10.1214/18-AOP1284ISI: 000466616100003OAI: oai:DiVA.org:kth-251704DiVA, id: diva2:1316836
Note

QC 20190521

Available from: 2019-05-21 Created: 2019-05-21 Last updated: 2019-05-21Bibliographically approved

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Schnelli, Kevin

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