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Combinatorial Properties of Brownian Motion on the Compact Classical Groups

Rains, E. M. (1997) Combinatorial Properties of Brownian Motion on the Compact Classical Groups. Journal of Theoretical Probability, 10 (3). pp. 659-679. ISSN 0894-9840. doi:10.1023/A:1022601711176.

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We consider the probability distribution on a classical group G which naturally generalizes the normal distribution (the “heat kernel”), defined in terms of Brownian motions on G. As Brownian motion can be defined in terms of the Laplacian on G, much of this work involves the computation of the Laplacian. These results are then used to study the behavior of the normal distribution on U(n) as n↦∞. In addition, we show how the results on computing the Laplacian on the classical groups can be used to compute expectations with respect to Haar measure on those groups.

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Additional Information:© 1997 Plenum Publishing Corporation. Received October 12, 1995; revised May 10, 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 Section 5 of Ref. 6; 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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AT&T Bell LaboratoriesUNSPECIFIED
Center for Communications Research (Princeton)UNSPECIFIED
Harvard UniversityUNSPECIFIED
Subject Keywords:Compact lie groups Brownian combinatorics
Issue or Number:3
Record Number:CaltechAUTHORS:20171106-161236257
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Official Citation:Rains, E.M. Journal of Theoretical Probability (1997) 10: 659.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83008
Deposited By: Tony Diaz
Deposited On:07 Nov 2017 02:44
Last Modified:15 Nov 2021 19:54

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