The Ising model on cubic maps: arbitrary genus
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912584968437760 |
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| author | Bousquet-Mélou, Mireille Carrance, Ariane Louf, Baptiste |
| author_facet | Bousquet-Mélou, Mireille Carrance, Ariane Louf, Baptiste |
| contents | We design a recursive algorithm to compute the partition function of the Ising model, summed over cubic maps with fixed size and genus. The algorithm runs in polynomial time, which is much faster than methods based on a Tutte-like, or topological, recursion.
We construct this algorithm out of a partial differential equation that we derive from the first equation of the KP hierarchy satisfied by the generating function of bipartite maps. This series is indeed related to the Ising partition function by a change of variables. We also obtain inequalities on the coefficients of this partition function, which should be useful for a probabilistic study of cubic Ising maps whose genus grows linearly with their size. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Ising model on cubic maps: arbitrary genus Bousquet-Mélou, Mireille Carrance, Ariane Louf, Baptiste Combinatorics Mathematical Physics 05A15 (Primary) 82B20, 37K10 (Secondary) We design a recursive algorithm to compute the partition function of the Ising model, summed over cubic maps with fixed size and genus. The algorithm runs in polynomial time, which is much faster than methods based on a Tutte-like, or topological, recursion. We construct this algorithm out of a partial differential equation that we derive from the first equation of the KP hierarchy satisfied by the generating function of bipartite maps. This series is indeed related to the Ising partition function by a change of variables. We also obtain inequalities on the coefficients of this partition function, which should be useful for a probabilistic study of cubic Ising maps whose genus grows linearly with their size. |
| title | The Ising model on cubic maps: arbitrary genus |
| topic | Combinatorics Mathematical Physics 05A15 (Primary) 82B20, 37K10 (Secondary) |
| url | https://arxiv.org/abs/2504.00768 |