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Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

Aiya Kuchukova, Marcus Pappik, Will Perkins, Corrine Yap

TL;DR

This work analyzes the Kawasaki dynamics for the fixed-magnetization Ising model on bounded-degree graphs, establishing fast mixing below the tree-uniqueness threshold and showing a regime where fast sampling and exponential Kawasaki mixing diverge.The authors develop a multifaceted approach combining $\ell_{\infty}$-spectral independence, Edgeworth-type local limit theorems, zero-freeness thresholds, and localization, together with metastability on random regular graphs to capture slow mixing phenomena.A key contribution is the precise separation between algorithmic and dynamical thresholds: even when efficient sampling exists via alternative methods, Kawasaki dynamics can remain exponentially slow in certain magnetization ranges, clarifying the limits of Markov-chain sampling in the fixed-magnetization setting.These results connect tree-uniqueness, analytic thresholds, and dynamical thresholds, contributing to a deeper understanding of the dynamics of spin systems on graphs and the complexity of sampling under fixed magnetization.

Abstract

We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree $Δ$. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree $Δ$ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.

Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

TL;DR

This work analyzes the Kawasaki dynamics for the fixed-magnetization Ising model on bounded-degree graphs, establishing fast mixing below the tree-uniqueness threshold and showing a regime where fast sampling and exponential Kawasaki mixing diverge.The authors develop a multifaceted approach combining $\ell_{\infty}$-spectral independence, Edgeworth-type local limit theorems, zero-freeness thresholds, and localization, together with metastability on random regular graphs to capture slow mixing phenomena.A key contribution is the precise separation between algorithmic and dynamical thresholds: even when efficient sampling exists via alternative methods, Kawasaki dynamics can remain exponentially slow in certain magnetization ranges, clarifying the limits of Markov-chain sampling in the fixed-magnetization setting.These results connect tree-uniqueness, analytic thresholds, and dynamical thresholds, contributing to a deeper understanding of the dynamics of spin systems on graphs and the complexity of sampling under fixed magnetization.

Abstract

We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree . Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.
Paper Structure (32 sections, 37 theorems, 182 equations, 5 figures)

This paper contains 32 sections, 37 theorems, 182 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview of Techniques
  3. Fast Mixing
  4. Slow Mixing
  5. Outline
  6. Preliminaries
  7. Ising Model on the Infinite Tree
  8. Pinned Models
  9. Kawasaki Dynamics, Down-up Walk, and Glauber Dynamics
  10. Mixing Times
  11. Upper Bounds on Mixing Time
  12. Lower Bounds on Mixing Time
  13. Thresholds for Zero-Freeness and Spectral Independence
  14. Rapid Mixing
  15. Edgeworth Expansion
  16. ...and 17 more sections

Key Result

Theorem 1.1

Fix $\Delta \geq 3, \beta \geq 0$, and $\eta \in [-1, 1]$. For the Kawasaki dynamics, the following two statements hold:

Figures (5)

  • Figure 1: Sketch of the phase space for the fixed-magnetization model on $\mathcal{G}_\Delta$ when $\Delta=4$, where $\bar{\eta_a} = \eta_{\Delta, \beta, \bar{\lambda}_a}$
  • Figure 2: Sketch of the phase space for the Ising model Glauber dynamics on $\mathcal{G}_{\Delta}$ when $\Delta=4$.
  • Figure 3: The structure of the rapid mixing proof
  • Figure 4: The structure of the slow mixing proof
  • Figure 5: Sketch of the function $f_{\Delta, \beta, \lambda}(\eta)$ for $\Delta = 4$, $\beta = \ln(2)+0.1$, and (left) $\lambda = 1.08$, (right) $\lambda = 1.01$.

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.4
  • Proposition 2.1
  • Theorem 2.2: CDKP21
  • Definition 2.3: Kawasaki dynamics
  • Definition 2.4: Down-up walk with plus pinnings
  • Definition 2.6: Glauber dynamics
  • Definition 2.7
  • Theorem 2.9: levin2017markov
  • ...and 70 more