Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mixing of general biased adjacent transposition chains

Reza Gheissari, Holden Lee, Eric Vigoda

TL;DR

This work analyzes a general biased adjacent transposition Markov chain on $S_n$ with pairwise biases $p_{i,j}$ and proves that whenever $p_{i,j}>\tfrac12+\varepsilon$ for all $i<j$, the mixing time is $\Theta(n^2)$ and the chain exhibits a pre-cutoff. The authors develop a multi-scale strategy: (i) a burn-in to $\ell$-localized configurations, (ii) spatial mixing within the localized set, and (iii) a recursive block-dynamics framework to obtain a polynomial bound, which is then boosted to the optimal quadratic bound. A key technical component is a spin-system-inspired spatial mixing result that holds on typical localized configurations, coupled with a domination-by-ASEP to control the burn-in. The paper also connects to ASEP/MKPZ-type behavior, providing a robust approach to polynomial mixing without monotonicity and offering insights into sampling and partition-function structure under general biases.

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 $Θ(n^2)$ 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.

Mixing of general biased adjacent transposition chains

TL;DR

This work analyzes a general biased adjacent transposition Markov chain on with pairwise biases and proves that whenever for all , the mixing time is and the chain exhibits a pre-cutoff. The authors develop a multi-scale strategy: (i) a burn-in to -localized configurations, (ii) spatial mixing within the localized set, and (iii) a recursive block-dynamics framework to obtain a polynomial bound, which is then boosted to the optimal quadratic bound. A key technical component is a spin-system-inspired spatial mixing result that holds on typical localized configurations, coupled with a domination-by-ASEP to control the burn-in. The paper also connects to ASEP/MKPZ-type behavior, providing a robust approach to polynomial mixing without monotonicity and offering insights into sampling and partition-function structure under general biases.

Abstract

We analyze the general biased adjacent transposition shuffle process, which is a well-studied Markov chain on the symmetric group . In each step, an adjacent pair of elements and are chosen, and then is placed ahead of with probability . 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 satisfying for all and a simple monotonicity condition, the mixing time is polynomial. We prove that for any fixed , as long as for all , the mixing time is 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.

Paper Structure

This paper contains 23 sections, 26 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Proof overview
  3. Burn-in to localized configurations
  4. Spatial mixing within the localized set
  5. Recursive argument to prove polynomial mixing
  6. Boosting to quadratic mixing time
  7. Spatial mixing for the stationary distribution
  8. Self-reducibility of the set of localized configurations
  9. Bounds on deviations of particle locations
  10. Typical configurations are logarithmically localized
  11. Spatial mixing for localized configurations
  12. Burn-in via dominating exclusion processes
  13. The asymmetric exclusion process
  14. Coupling the biased adjacent transposition shuffle and ASEP
  15. Analysis of the ASEP
  16. ...and 8 more sections

Key Result

Theorem 1.1

For every $\varepsilon>0$ and any fixed $\delta>0$, there exists $C(\varepsilon)$ such that for any $(p_{i,j})_{1\le i< j\le n}$ which satisfies $\varepsilon$-positive bias, the adjacent transposition Markov chain $\mathcal{M}_{\mathsf{AT}}$ has mixing timeHere we are using $\widetilde{O}$ to mean u In particular, the family has pre-cutoff (see LP): there exists $C(\varepsilon)$ such that for any

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • ...and 48 more