Mixing of general biased adjacent transposition chains

TL;DR

This paper proves that the biased adjacent transposition Markov chain on permutations mixes in quadratic time when biases are sufficiently strong, using novel spatial mixing and multiscale techniques.

Contribution

It establishes polynomial mixing time for general biased transposition chains with biases above 1/2, extending previous conjectures and applying new spatial mixing methods.

Findings

Mixing time is Θ(n^2) for biases > 1/2 + ε.

Chain exhibits pre-cutoff behavior.

Introduces spatial mixing analysis after burn-in period.

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.