Mixing Time for Some Adjacent Transposition Markov Chains

Abstract: We prove rapid mixing for certain Markov chains on the set $S_n$ of permutations on $1,2,\dots,n$ in which adjacent transpositions are made with probabilities that depend on the items being transposed. Typically, when in state $\sigma$, a position $i<n$ is chosen uniformly at random, and $\sigma(i)$ and $\sigma(i{+}1)$ are swapped with probability depending on $\sigma(i)$ and $\sigma(i{+}1)$. The stationary distributions of such chains appear in various fields of theoretical computer science, and rapid mixing established in the uniform case. Recently, there has been progress in cases with biased stationary distributions, but there are wide classes of such chains whose mixing time is unknown. One case of particular interest is what we call the "gladiator chain," in which each number $g$ is assigned a "strength" $s_g$ and when $g$ and $g'$ are adjacent and chosen for possible swapping, $g$ comes out on top with probability $s_g/(s_g + s_{g'})$. We obtain a polynomial-time upper bound on mixing time when the gladiators fall into only three strength classes. A preliminary version of this paper appeared as "Mixing of Permutations by Biased Transposition" in STACS 2017.