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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.56
Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

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

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

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.
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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)

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@InProceedings{kuchukova_et_al:LIPIcs.APPROX/RANDOM.2024.56,
  author =	{Kuchukova, Aiya and Pappik, Marcus and Perkins, Will and Yap, Corrine},
  title =	{{Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{56:1--56:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.56},
  URN =		{urn:nbn:de:0030-drops-210493},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.56},
  annote =	{Keywords: ferromagnetic Ising model, fixed-magnetization Ising model, Kawasaki dynamics, Glauber dynamics, mixing time}
}
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