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arXiv · 2405.06209

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

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.

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Aiya Kuchukova, Marcus Pappik, Will Perkins, Corrine Yap. 2024-05-10. Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs. https://doi.org/10.1002/rsa.70038

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