Edge-Tilting Field Dynamics: Rapid Mixing at the Uniqueness Threshold and Optimal Mixing for Swendsen-Wang Dynamics
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911637413298176 |
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| author | Chen, Xiaoyu Ju, Zhe Miao, Tianshun Yin, Yitong Zhang, Xinyuan |
| author_facet | Chen, Xiaoyu Ju, Zhe Miao, Tianshun Yin, Yitong Zhang, Xinyuan |
| contents | We prove two results on the mixing times of Markov chains for two-spin systems. First, we show that the Glauber dynamics mixes in polynomial time for the Gibbs distributions of antiferromagnetic two-spin systems at the critical threshold of the uniqueness phase transition of the Gibbs measure on infinite regular trees. This completes the computational phase transition picture for antiferromagnetic two-spin systems, which includes near-linear-time optimal mixing in the uniqueness regime [Chen--Liu--Vigoda, STOC '21; Chen--Feng--Yin--Zhang, FOCS '22], NP-hardness of approximate sampling in the non-uniqueness regime [Sly--Sun, FOCS '12], and polynomial-time mixing at criticality (this work).
Second, we prove an optimal $O(\log n)$ mixing time bound as well as an optimal $Ω(1)$ spectral gap for the Swendsen--Wang dynamics for the ferromagnetic Ising model with an external field on bounded-degree graphs. To the best of our knowledge, these are the first sharp bounds on the mixing rate of this classical global Markov chain beyond mean-field or strong spatial mixing (SSM) regimes, and resolve a conjecture of [Feng--Guo--Wang, IANDC '23].
A key ingredient in both proofs is a new family of localization schemes that extends the field dynamics of [Chen--Feng--Yin--Zhang, FOCS '21] by tilting general edge (or hyperedge) weights rather than vertex fields. This framework, which subsumes the classical Swendsen--Wang dynamics as a special case, extends the localization framework of [Chen--Eldan, FOCS '22] beyond stochastic and field localizations, and enables controlled tilting of interaction strengths while preserving external fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10525 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Edge-Tilting Field Dynamics: Rapid Mixing at the Uniqueness Threshold and Optimal Mixing for Swendsen-Wang Dynamics Chen, Xiaoyu Ju, Zhe Miao, Tianshun Yin, Yitong Zhang, Xinyuan Data Structures and Algorithms Probability We prove two results on the mixing times of Markov chains for two-spin systems. First, we show that the Glauber dynamics mixes in polynomial time for the Gibbs distributions of antiferromagnetic two-spin systems at the critical threshold of the uniqueness phase transition of the Gibbs measure on infinite regular trees. This completes the computational phase transition picture for antiferromagnetic two-spin systems, which includes near-linear-time optimal mixing in the uniqueness regime [Chen--Liu--Vigoda, STOC '21; Chen--Feng--Yin--Zhang, FOCS '22], NP-hardness of approximate sampling in the non-uniqueness regime [Sly--Sun, FOCS '12], and polynomial-time mixing at criticality (this work). Second, we prove an optimal $O(\log n)$ mixing time bound as well as an optimal $Ω(1)$ spectral gap for the Swendsen--Wang dynamics for the ferromagnetic Ising model with an external field on bounded-degree graphs. To the best of our knowledge, these are the first sharp bounds on the mixing rate of this classical global Markov chain beyond mean-field or strong spatial mixing (SSM) regimes, and resolve a conjecture of [Feng--Guo--Wang, IANDC '23]. A key ingredient in both proofs is a new family of localization schemes that extends the field dynamics of [Chen--Feng--Yin--Zhang, FOCS '21] by tilting general edge (or hyperedge) weights rather than vertex fields. This framework, which subsumes the classical Swendsen--Wang dynamics as a special case, extends the localization framework of [Chen--Eldan, FOCS '22] beyond stochastic and field localizations, and enables controlled tilting of interaction strengths while preserving external fields. |
| title | Edge-Tilting Field Dynamics: Rapid Mixing at the Uniqueness Threshold and Optimal Mixing for Swendsen-Wang Dynamics |
| topic | Data Structures and Algorithms Probability |
| url | https://arxiv.org/abs/2604.10525 |