Random Walks on Dihedral Groups

Joseph McCollum ()
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Joseph McCollum: Siena College

Journal of Theoretical Probability, 2011, vol. 24, issue 2, 397-408

Abstract: 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 .

Keywords: Random walk; Dihedral group; Lower bounds; 60G50 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-010-0307-6

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