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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2312.16911 |
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| _version_ | 1866915592617852928 |
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| author | Taggi, Lorenzo Wu, Wei |
| author_facet | Taggi, Lorenzo Wu, Wei |
| contents | We study a generalisation of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, spatial random permutations, the dimer model, and the spin $O(N)$ model, and which is also related to the loop $O(N)$ model. We show that on two-dimensional-like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices decay to zero as the distance between the vertices diverges. Our approach is based on the introduction of a new complex spin representation for all models in this class, together with a new proof of the Mermin-Wagner theorem that does not require positivity of the Gibbs measure. Even for the well-studied dimer and double-dimer models our results are new: since they do not rely on exact solvability or Kasteleyn's theorem, they apply beyond the planar-graph setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16911 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mermin-Wagner theorem for dimers, monomer double-dimers, and spatial random permutations Taggi, Lorenzo Wu, Wei Probability 82B20, 05C70, 82B26 We study a generalisation of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, spatial random permutations, the dimer model, and the spin $O(N)$ model, and which is also related to the loop $O(N)$ model. We show that on two-dimensional-like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices decay to zero as the distance between the vertices diverges. Our approach is based on the introduction of a new complex spin representation for all models in this class, together with a new proof of the Mermin-Wagner theorem that does not require positivity of the Gibbs measure. Even for the well-studied dimer and double-dimer models our results are new: since they do not rely on exact solvability or Kasteleyn's theorem, they apply beyond the planar-graph setting. |
| title | Mermin-Wagner theorem for dimers, monomer double-dimers, and spatial random permutations |
| topic | Probability 82B20, 05C70, 82B26 |
| url | https://arxiv.org/abs/2312.16911 |