Embedding a Markov Chain into a Random Walk on a Permutation Group

  • Authors:

  • Steven N. Evans

  • Affiliations:

  • Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, CA 94720-3860, USA (e-mail: evans@stat.berkeley.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2004

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Abstract

Using representation theory, we obtain a necessary and sufficient condition for a discrete-time Markov chain on a finite state space $E$ to be representable as $$\Psi_n \Psi_{n-1} \cdots \Psi_1 z,\quad n \geq 0,$$ for any $z \in E$, where the $\Psi_i$ are independent, identically distributed random permutations taking values in some given transitive group of permutations on $E$. The condition is particularly simple when the group is 2-transitive on $E$. We also work out the explicit form of our condition for the dihedral group of symmetries of a regular polygon.