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Combinatorial Properties of Brownian Motion on the Compact Classical Groups

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  • E. M. Rains

Abstract

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 $$n \mapsto \infty $$ . 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.

Suggested Citation

  • E. M. Rains, 1997. "Combinatorial Properties of Brownian Motion on the Compact Classical Groups," Journal of Theoretical Probability, Springer, vol. 10(3), pages 659-679, July.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022601711176
    DOI: 10.1023/A:1022601711176
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    Cited by:

    1. Ping Zhong, 2015. "On the Free Convolution with a Free Multiplicative Analogue of the Normal Distribution," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1354-1379, December.
    2. Todd Kemp, 2017. "Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N," Journal of Theoretical Probability, Springer, vol. 30(2), pages 397-451, June.
    3. Luc Deleaval & Nizar Demni, 2018. "Moments of the Hermitian Matrix Jacobi Process," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1759-1778, September.

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