Mixing of general biased adjacent transposition chains

Abstract

We analyze the general biased adjacent transposition shuffle process, which is a well-studied Markov chain on the symmetric group $S_n$. In each step, an adjacent pair of elements $i$ and $j$ are chosen, and then $i$ is placed ahead of $j$ with probability $p_{ij}$. This Markov chain arises in the study of self-organizing lists in theoretical computer science, and has close connections to exclusion processes from statistical physics and probability theory. Fill (2003) conjectured that for general $p_{ij}$ satisfying $p_{ij} \ge 1/2$ for all $i<j$ and a simple monotonicity condition, the mixing time is polynomial. We prove that for any fixed $\varepsilon>0$, as long as $p_{ij} >1/2+\varepsilon$ for all $i<j$, the mixing time is $\Theta(n^2)$ and exhibits pre-cutoff. Our key technical result is a form of spatial mixing for the general biased transposition chain after a suitable burn-in period. In order to use this for a mixing time bound, we adapt multiscale arguments for mixing times from the setting of spin systems to the symmetric group.