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Semigroups, Rings, and Markov Chains

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  • Kenneth S. Brown

    (Cornell University)

Abstract

We analyze random walks on a class of semigroups called “left-regular bands.” These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are “generalized derangement numbers,” which may be of independent interest.

Suggested Citation

  • Kenneth S. Brown, 2000. "Semigroups, Rings, and Markov Chains," Journal of Theoretical Probability, Springer, vol. 13(3), pages 871-938, July.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:3:d:10.1023_a:1007822931408
    DOI: 10.1023/A:1007822931408
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    Cited by:

    1. C. Y. Amy Pang, 2019. "Lumpings of Algebraic Markov Chains Arise from Subquotients," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1804-1844, December.

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