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

TL;DR

This paper analyzes the mixing times of Kawasaki dynamics for the fixed-magnetization Ising model on bounded-degree graphs, establishing conditions for rapid mixing and identifying regimes where it is slow, thus clarifying the model's computational complexity.

Contribution

It proves the conjecture that Kawasaki dynamics mix rapidly below the tree uniqueness threshold and shows that fast mixing does not hold in all tractable regimes, revealing nuanced behavior.

Findings

Kawasaki dynamics mix rapidly below the tree uniqueness threshold.

There exist parameter regimes where sampling is efficient but Kawasaki dynamics mix slowly.

Spectral independence and metastability thresholds are key tools in analysis.

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 $\Delta$. 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 $\Delta$ 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.