Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916977656725504 |
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| author | Hansen, Ulrik Thinggaard Klausen, Frederik Ravn Wildemann, Peter |
| author_facet | Hansen, Ulrik Thinggaard Klausen, Frederik Ravn Wildemann, Peter |
| contents | We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs Hansen, Ulrik Thinggaard Klausen, Frederik Ravn Wildemann, Peter Probability Mathematical Physics 82B05 82B20, 82B26, 82B43, 05C80, 60K35 We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide. |
| title | Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs |
| topic | Probability Mathematical Physics 82B05 82B20, 82B26, 82B43, 05C80, 60K35 |
| url | https://arxiv.org/abs/2410.22061 |