The single ring theorem

Guionnet, Alice and Krishnapur, Manjunath and Zeitouni, Ofer (2011) The single ring theorem. In: Annals of Mathematics, 174 (2). pp. 1189-1217.

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Abstract

We study the empirical measure LA of the eigenvalues of nonnormal square matrices of the form A(n) = U(n)T(n)V(n), with U(n), V(n) independent Haar distributed on the unitary group and T(n) diagonal. We show that when the empirical measure of the eigenyalues of T(n) converges, and T(n) satisfies some technical conditions, L(An) converges towards a rotationally invariant measure mu on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [6]. We also consider the case where U(n), V(n) are independently Haar distributed on the orthogonal group.

Item Type:	Journal Article
Publication:	Annals of Mathematics
Publisher:	John Hopkins University Press
Additional Information:	Copyright of this article belongs to John Hopkins University Press.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	04 Nov 2011 09:54
Last Modified:	04 Nov 2011 09:54
URI:	http://eprints.iisc.ac.in/id/eprint/41835

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