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November 2017 Local single ring theorem
Florent Benaych-Georges
Ann. Probab. 45(6A): 3850-3885 (November 2017). DOI: 10.1214/16-AOP1151


The single ring theorem, by Guionnet, Krishnapur and Zeitouni in Ann. of Math. (2) 174 (2011) 1189–1217, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, that is, an $N\times N$ matrix of the form $A=UTV$, with $U,V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale $(\log N)^{-1/4}$. On our way to prove it, we prove a matrix subordination result for singular values of sums of non-Hermitian matrices, as Kargin did in Ann. Probab. 43 (2015) 2119–2150 for Hermitian matrices. This allows to prove a local law for the singular values of the sum of two non-Hermitian matrices and a delocalization result for singular vectors.


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Florent Benaych-Georges. "Local single ring theorem." Ann. Probab. 45 (6A) 3850 - 3885, November 2017.


Received: 1 February 2015; Revised: 1 April 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 1383.15033
MathSciNet: MR3729617
Digital Object Identifier: 10.1214/16-AOP1151

Primary: 15B52 , 46L54 , 60B20

Keywords: Free convolution , free probability theory , Haar measure , local laws , random matrices , single ring theorem

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 6A • November 2017
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