On Transience of Card Shuffling

We present simple proofs of transience/recurrence for certain card shuffling models, that is, random walks on the infinite symmetric group. 1. Card shuffling models In this note, we consider several models of shuffling an infinite deck of cards. One of these models has been considered previously by Lawler [La]; our methods (using flows, shorting and comparison of Dirichlet forms) will – partially – simplify his result. Card shuffling is formalized by performing successive i.i.d. random permutations in the group S∞ of all permutations of the positive integers N that leave all but finitely many elements fixed. We identify the symmetric group Sn with the subgroup of S∞ fixing all elements > n, so that S∞ is the union of the Sn. We read the product of two permutations x, y from left to right, that is, x · y sends j ∈ N to y ( x(j) ) . Let μ be a symmetric probability measure on S∞ whose support generates the whole group. Associate with it a sequence Xn, n ≥ 1, of i.i.d. S∞-valued random variables with common distribution μ, and consider the associated random walk Zn = X1 · · ·Xn. This means that we start with the deck of cards in order (Z0 is the identity), and at each step we choose a random permutation Xn according to μ which tells us how to shuffle once more what we had obtained previously. The question addressed here is the following. Will the deck of cards eventually return to its original order with probability one (recurrence), or is this probability strictly smaller than 1 (transience)? We now describe four shuffling models, that is, probabilities μ1, . . . , μ4, each one governing another random walk. We start by considering a probability distribution p(·) on {2, 3, . . .}. 1) First choose n with probability p(n) and then j ∈ {1, . . . , n− 1} with probability 1/(n− 1), and exchange the n-th with the j-th card. Writing t(n, j) for the transposition of n and j, we have