The Annals of Probability

Local single ring theorem

Florent Benaych-Georges

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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.

Article information

Ann. Probab., Volume 45, Number 6A (2017), 3850-3885.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15B52: Random matrices 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 46L54: Free probability and free operator algebras

Random matrices single ring theorem local laws free convolution free probability theory haar measure


Benaych-Georges, Florent. Local single ring theorem. Ann. Probab. 45 (2017), no. 6A, 3850--3885. doi:10.1214/16-AOP1151.

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