Journal of Theoretical Probability

, Volume 13, Issue 3, pp 871–938

Semigroups, Rings, and Markov Chains

  • Kenneth S. Brown

DOI: 10.1023/A:1007822931408

Cite this article as:
Brown, K.S. Journal of Theoretical Probability (2000) 13: 871. doi:10.1023/A:1007822931408


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.

random walk Markov chain semigroup hyperplane arrangement diagonalization matroid derangement number 

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Kenneth S. Brown
    • 1
  1. 1.Department of MathematicsCornell UniversityIthaca

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