Georgia Institute of Technology College of Sciences

Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain M_n is known to converge in time \Theta(n^3 \log n) in the uniform case and time \Theta(n^2) in the constant bias case, in which we put adjacent elements in order with probability p \neq 1/2 and out of order with probability 1-p. In joint work with Prateek Bhakta, Dana Randall and Amanda Streib, we consider the variable bias case where the probability of putting an adjacent pair of elements in order depends on the two elements, and we put adjacent elements x < y in order with probability p_{x,y} and out of order with probability 1-p_{x,y}. The problem of bounding the mixing rate of M_n was posed by Fill and was motivated by the Move-Ahead-One self-organizing list update algorithm. It was conjectured that the chain would always be rapidly mixing if 1/2 \leq p_{x,y} \leq 1 for all x < y, but this was only known in the case of constant bias or when p_{x,y} is equal to 1/2 or 1, a case that corresponds to sampling linear extensions of a partial order. We prove the chain is rapidly mixing for two classes: ``Choose Your Weapon,'' where we are given r_1,..., r_{n-1} with r_i \geq 1/2 and p_{x,y}=r_x for all x < y (so the dominant player chooses the game, thus fixing his or her probability of winning), and ``League Hierarchies,'' where there are two leagues and players from the A-league have a fixed probability of beating players from the B-league, players within each league are similarly divided into sub-leagues with a possibly different fixed probability, and so forth recursively. Both of these classes include permutations with constant bias as a special case. Moreover, we also prove that the most general conjecture is false. We do so by constructing a counterexample where 1/2 \leq p_{x,y} \leq 1 for all x < y, but for which the nearest neighbor transposition chain requires exponential time to converge.