A General Coupling for Ising Models and Beyond
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918057080782848 |
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| author | Hansen, Ulrik Thinggaard Jiang, Jianping Klausen, Frederik Ravn |
| author_facet | Hansen, Ulrik Thinggaard Jiang, Jianping Klausen, Frederik Ravn |
| contents | The couplings between the Ising model and its graphical representations, the random-cluster, random current and loop $\mathrm{O}(1)$ models, are put on common footing through a generalization of the Swendsen-Wang-Edwards-Sokal coupling. A new special case is that the single random current with parameter $2J$ on edges of agreement of XOR-Ising spins has the law of the double random current. The coupling also yields a general mechanism for constructing conditional Bernoulli percolation measures via uniform sampling, providing new perspectives on models such as the arboreal gas, random $d$-regular graphs, and self-avoiding walks. As an application of the coupling for the Loop-Cluster joint model, we prove that the regime of exponential decay of the $q$-state Potts model, the random-cluster representation, and its $q$-flow loop representation coincides on the torus, generalizing the Ising result. Finally, the Loop-Cluster coupling is generalized to lattice gauge theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A General Coupling for Ising Models and Beyond Hansen, Ulrik Thinggaard Jiang, Jianping Klausen, Frederik Ravn Probability Mathematical Physics 82B05 82B20, 82B26, 82B43, 05C80, 60K35 The couplings between the Ising model and its graphical representations, the random-cluster, random current and loop $\mathrm{O}(1)$ models, are put on common footing through a generalization of the Swendsen-Wang-Edwards-Sokal coupling. A new special case is that the single random current with parameter $2J$ on edges of agreement of XOR-Ising spins has the law of the double random current. The coupling also yields a general mechanism for constructing conditional Bernoulli percolation measures via uniform sampling, providing new perspectives on models such as the arboreal gas, random $d$-regular graphs, and self-avoiding walks. As an application of the coupling for the Loop-Cluster joint model, we prove that the regime of exponential decay of the $q$-state Potts model, the random-cluster representation, and its $q$-flow loop representation coincides on the torus, generalizing the Ising result. Finally, the Loop-Cluster coupling is generalized to lattice gauge theories. |
| title | A General Coupling for Ising Models and Beyond |
| topic | Probability Mathematical Physics 82B05 82B20, 82B26, 82B43, 05C80, 60K35 |
| url | https://arxiv.org/abs/2506.10765 |