Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations

We study the mixing time of a Markov chain Mnn on permutations that performs nearest neighbor transpositions in the non-uniform setting, a problem arising in the context of self-organizing lists. We are given “positively biased” probabilities {pi,j ≥ 1/2} for all i < j and let pj,i = 1− pi,j . In each step, the chain Mnn chooses two adjacent elements k, and ` and exchanges their positions with probability p`,k. Here we define two general classes and give the first proofs that the chain is rapidly mixing for both. In the first case we are given constants r1, . . . rn−1 with 1/2 ≤ ri ≤ 1 for all i and we set pi,j = ri for all i < j. In the second we are given a binary tree with n leaves labeled 1, . . . n and constants q1, . . . qn−1 associated with all of the internal vertices, and we let pi,j = qi∧j for all i < j. Our bounds on the mixing time of Mnn rely on bijections between permutations, inversion tables and asymmetric simple exclusion processes (ASEPs) that allow us to express moves of the chain in the context of these other combinatorial families. We also demonstrate that the chain is not always rapidly mixing by constructing an example requiring exponential time to converge to equilibrium. This proof relies on a reduction to biased lattice paths in Z.