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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On connections between the the theory of random operators and the theory of random matrices


Author: L. Pastur
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 1.
Journal: St. Petersburg Math. J. 23 (2012), 117-137
MSC (2010): Primary 15B52, 60B20, 47B80
Published electronically: November 8, 2011
MathSciNet review: 2760151
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Abstract | References | Similar Articles | Additional Information

Abstract: For several families of selfadjoint ergodic operators, it is proved that, as the parameter that indexes the operators of a family tends to infinity, the integrated density of states converges weakly to the infinite size limit of the normalized counting measure of eigenvalues of certain random matrices. The subsequent informal discussion is devoted to the role of these results as possible indications of the presence of the continuous spectrum for random ergodic operators belonging to the families under consideration, when the indexing parameter values are sufficiently large.


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Additional Information

L. Pastur
Affiliation: Mathematics Division, Institute for Low Temperatures, Kharkov, Ukraine
Email: pastur2001@yahoo.com

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01189-6
PII: S 1061-0022(2011)01189-6
Keywords: Random matrices, random ergodic operators, integrated density of states
Received by editor(s): September 15, 2010
Published electronically: November 8, 2011
Dedicated: To the memory of M. Sh. Birman
Article copyright: © Copyright 2011 American Mathematical Society