A Markov Chain Structure for Riff Shuffling

Abstract

A Markov chain structure can be exploited to study card shuffling problems involving the riff shuffle, i.e., the shuffle which cuts the deck of cards into two parts and then recombines those parts by interleaving the cards. The Markov chain structure is imposed by regarding the different arrangements of the deck as states of a chain, and assigning transition probabilities in accordance with a probability distribution on the set of shuffles. We confine our study to decks with an even number n of cards, and to riff shuffles which cut the deck at the center. A lexicographical scheme for indexing the states of the chain is developed ; by regarding the riff shuffles as a subset of the symmetric group of degree n, it is verified that the transition probability matrix relative to this indexing is centrosymmetric ; this fact greatly facilitates the computation of eigenvalues for the matrix, the knowledge of which in turn permits resolution of certain questions concerning the Markov chain. The Markov chain is also shown to be collapsible by introducing a subgroup of the symmetric group of degree n, each element of which commutes with all riff shuffles ; and it is shown that the eigenvalues of the transition probability matrix of the collapsed chain are also eigenvalues of the original transition probability matrix. Finally, some observations concerning the possibility of identifying the collapsed Markov chain with a shuffling problem of lower dimension are made. This paper leads to several interesting and unresolved questions: e.g., the physical significance of the comparison of the set of eigenvalues of the transition probability matrix of the collapsed Markov chain, and the set of eigenvalues of the transition probability matrix of the original Markov chain, and also the question of whether or not the collapsed chain is in general identifiable with a shuffling problem of lower dimension and what possible recursive relations might result.