Mixing of general biased adjacent transposition chains

2025-11-04

AI Summary

This work analyzes the mixing time of generalized biased adjacent transposition Markov chains on the symmetric group. For the strongly biased regime where transition probabilities satisfy $p > frac{1}{2} + varepsilon$, we establish that the mixing time is $Theta(n^2)$ and exhibits cutoff—a first such result for permutation groups—by developing a novel multi-scale analytical framework. Our method integrates coupling techniques for Markov chains, spatial mixing arguments, scale-recursion tools adapted from spin systems, and post-burn-in dynamic evolution estimates. The contributions include: (i) tight polynomial bounds on mixing time; (ii) rigorous confirmation of Fill’s conjecture under strong bias; and (iii) uncovering a structural correspondence between self-organizing lists and exclusion processes. These results advance the theoretical understanding of biased permutation dynamics and provide new analytical machinery for studying non-reversible, high-dimensional Markov chains with algebraic structure.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

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.

Problem

• Analyzing mixing time of biased adjacent transposition Markov chains
• Proving polynomial mixing time under specific probability conditions
• Establishing spatial mixing properties for transposition processes

Innovation

• Biased adjacent transposition Markov chain
• Spatial mixing after burn-in period
• Multiscale arguments adapted from spin systems