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Random Walks on Dihedral Groups

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  • Joseph McCollum

    (Siena College)

Abstract

In this paper, we look at the lower bounds of two specific random walks on the dihedral group. The first theorem discusses a random walk generated with equal probabilities by one rotation and one flip. We show that roughly p 2 steps are necessary for the walk to become close to uniformly distributed on all of D 2p where p≥3 is an integer. Next we take a random walk on the dihedral group generated by a random k-subset of the dihedral group. The latter theorem shows that it is necessary to take roughly p 2/(k−1) steps in the typical random walk to become close to uniformly distributed on all of D 2p . We note that there is at least one rotation and one flip in the k-subset, or the random walk generated by this subset has periodicity problems or will not generate all of D 2p .

Suggested Citation

  • Joseph McCollum, 2011. "Random Walks on Dihedral Groups," Journal of Theoretical Probability, Springer, vol. 24(2), pages 397-408, June.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:2:d:10.1007_s10959-010-0307-6
    DOI: 10.1007/s10959-010-0307-6
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