Geodesic
The single ring theorem
Alice Guionnet1 ; Manjunath Krishnapur2 ; Ofer Zeitouni3
1 UMPA<br/> Ecole Normale Supérieure de Lyon<br/> 69264 Lyon<br/> France
2 Indian Institute of Science<br/>Bangalore<br/> India
3 Department of Mathematics<br/> University of Minnesota<br/> Minneapolis, MN 55455<br/><br/> and<br/><br/> Weizmann Institute of Science,<br/>Rehovot 76100<br/> Israel
Annals of mathematics, Tome 174 (2011) no. 2, pp. 1189-1217.

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We study the empirical measure $L_{A_n}$ of the eigenvalues of nonnormal square matrices of the form $A_n=U_nT_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $T_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $T_n$ converges, and $T_n$ satisfies some technical conditions, $L_{A_n}$ 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 [FZ]. We also consider the case where $U_n,V_n$ are independently Haar distributed on the orthogonal group.
DOI : 10.4007/annals.2011.174.2.10

Alice Guionnet 1 ; Manjunath Krishnapur 2 ; Ofer Zeitouni 3

1 UMPA<br/> Ecole Normale Supérieure de Lyon<br/> 69264 Lyon<br/> France
2 Indian Institute of Science<br/>Bangalore<br/> India
3 Department of Mathematics<br/> University of Minnesota<br/> Minneapolis, MN 55455<br/><br/> and<br/><br/> Weizmann Institute of Science,<br/>Rehovot 76100<br/> Israel 
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     title = {The single ring theorem},
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     pages = {1189--1217},
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Alice Guionnet; Manjunath Krishnapur; Ofer Zeitouni. The single ring theorem. Annals of mathematics, Tome 174 (2011) no. 2, pp. 1189-1217. doi : 10.4007/annals.2011.174.2.10. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.2.10/

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