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Computer Science > Data Structures and Algorithms

arXiv:2405.06209 (cs)
[Submitted on 10 May 2024]

Title: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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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.
Subjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO); Probability (math.PR)
MSC classes: Primary: 60J10, Secondary: 82B20, 68W20
Cite as: arXiv:2405.06209 [cs.DS]
  (or arXiv:2405.06209v1 [cs.DS] for this version)
  https://doi.org/10.48550/arXiv.2405.06209

Submission history

From: Corrine Yap [view email]
[v1] Fri, 10 May 2024 02:45:54 UTC (104 KB)
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