Rains, E. M. (1997) High powers of random elements of compact Lie groups. Probability Theory and Related Fields, 107 (2). pp. 219-241. ISSN 0178-8051. doi:10.1007/s004400050084. https://resolver.caltech.edu/CaltechAUTHORS:20171107-075059031
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Abstract
If a random unitary matrix U 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 U falling in a given arc, as the dimension of U 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].
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| Additional Information: | © 1997 Springer-Verlag Berlin Heidelberg. Received: 15 October 1995; In revised form: 7 March 1996. The greatest thanks are due to Persi Diaconis (the author’s thesis advisor), for his many suggestions of problems (a small sampling of which appear in the present work), for his helpful advice when it came time to revise what had been written, and for his encouragement of the author’s procrastination. Thanks are also due to the following people and institutions, in no particular order: Andrew Odlyzko, for many helpful comments on Sect. 5 of [9]; AT&T Bell Laboratories (Murray Hill) and the Center for Communications Research (Princeton) for generous summer support; the Harvard University Mathematics Department and the National Science Foundation, for generous support for the rest of the year. Also, thanks are due, for helpful comments, to Nantel Bergeron, Maurice Rojas, Richard Stanley, and Dan Stroock. Last, but not least, the author owes thanks to Wojbor Woyczynski, both for introducing him to probability theory and for introducing him to Prof. Diaconis; the thesis excerpted here would be very different, had either introduction not been made. | ||||||||||||
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| Issue or Number: | 2 | ||||||||||||
| Classification Code: | Mathematics Subject Classification: (1991): 60B15, 22E99 | ||||||||||||
| DOI: | 10.1007/s004400050084 | ||||||||||||
| Record Number: | CaltechAUTHORS:20171107-075059031 | ||||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20171107-075059031 | ||||||||||||
| Official Citation: | Rains, E. Probab Theory Relat Fields (1997) 107: 219. https://doi.org/10.1007/s004400050084 | ||||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
| ID Code: | 83015 | ||||||||||||
| Collection: | CaltechAUTHORS | ||||||||||||
| Deposited By: | Tony Diaz | ||||||||||||
| Deposited On: | 07 Nov 2017 15:56 | ||||||||||||
| Last Modified: | 15 Nov 2021 19:54 |
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