Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911281191059456 |
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| author | Kuchukova, Aiya Pappik, Marcus Perkins, Will Yap, Corrine |
| author_facet | Kuchukova, Aiya Pappik, Marcus Perkins, Will Yap, Corrine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs Kuchukova, Aiya Pappik, Marcus Perkins, Will Yap, Corrine Data Structures and Algorithms Combinatorics Probability Primary: 60J10, Secondary: 82B20, 68W20 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. |
| title | Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs |
| topic | Data Structures and Algorithms Combinatorics Probability Primary: 60J10, Secondary: 82B20, 68W20 |
| url | https://arxiv.org/abs/2405.06209 |