On counting polygons in a crystal
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917182821105664 |
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| author | Grimmett, Geoffrey R. |
| author_facet | Grimmett, Geoffrey R. |
| contents | How many $n$-step polygons exist that contain a given vertex of an infinite quasi-transitive graph $G$? The exponential growth rate of such polygons is identified as the connective constant when $G$ has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that $G$ has a certain ${\Bbb Z}^2$ action of automorphisms. The main theorem extends a result of Hammersley (Proc. Cambridge Philos. Soc. 57 (1961) 516--523) and others for the hypercubic lattice, and responds to Hammersley's challenge to prove such a result for more general "crystals''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01128 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On counting polygons in a crystal Grimmett, Geoffrey R. Combinatorics Probability 05C30, 05C38, 60K35, 82B20 How many $n$-step polygons exist that contain a given vertex of an infinite quasi-transitive graph $G$? The exponential growth rate of such polygons is identified as the connective constant when $G$ has sub-exponential growth and possesses a so-called square graph height function. The last condition amounts to the requirement that $G$ has a certain ${\Bbb Z}^2$ action of automorphisms. The main theorem extends a result of Hammersley (Proc. Cambridge Philos. Soc. 57 (1961) 516--523) and others for the hypercubic lattice, and responds to Hammersley's challenge to prove such a result for more general "crystals''. |
| title | On counting polygons in a crystal |
| topic | Combinatorics Probability 05C30, 05C38, 60K35, 82B20 |
| url | https://arxiv.org/abs/2601.01128 |