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

Authors Aiya Kuchukova , Marcus Pappik , Will Perkins , Corrine Yap

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Author Details

Aiya Kuchukova
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Marcus Pappik
  • Hasso Plattner Institute, University of Potsdam, Germany
Will Perkins
  • School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA
Corrine Yap
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

Funding

  • Pappik, Marcus: supported by the HPI Research School on Data Science and Engineering.
  • Perkins, Will: supported in part by NSF grant CCF-2309708.
  • Yap, Corrine: supported in part by NIH grant R01GM126554.

Acknowledgements

The authors thank Zongchen Chen and Marcus Michelen for very helpful discussions.

Cite AsGet BibTex

Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 56:1-56:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.56

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov-chain Monte Carlo methods
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • ferromagnetic Ising model
  • fixed-magnetization Ising model
  • Kawasaki dynamics
  • Glauber dynamics
  • mixing time

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