High powers of random elements of compact Lie groups

If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to...

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Bibliographic Details
Published in:Probability theory and related fields 1997-02, Vol.107 (2), p.219-241
Main Author: RAINS, E. M
Format: Article
Language:English
Subjects:
Eigenvalues
Exact sciences and technology
Group theory
Lie groups
Mathematics
Probability
Probability and statistics
Probability theory and stochastic processes
Probability theory on algebraic and topological structures
Sciences and techniques of general use
Topological groups, lie groups
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Description
Summary:If a random unitary matrix is raised to a sufficiently high power, its eigenvalues are exactly distributed as independent, uniform phases. We prove this result, and apply it to give exact asymptotics of the variance of the number of eigenvalues of falling in a given arc, as the dimension of tends to infinity. The independence result, it turns out, extends to arbitrary representations of arbitrary compact Lie groups. We state and prove this more general theorem, paying special attention to the compact classical groups and to wreath products. This paper is excerpted from the author's doctoral thesis, [9].
ISSN:0178-8051
1432-2064
DOI:10.1007/s004400050084