Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs
<|reference_start|>Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs: 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.<|reference_end|>
@article{kuchukova2024fast,
    title={Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs},
    author={Aiya Kuchukova, Marcus Pappik, Will Perkins, Corrine Yap},
    journal={arXiv preprint arXiv:2405.06209},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.06209},
    primaryClass={cs.DS math.CO math.PR}
}