Spanning forests in regular planar maps
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2013
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| _version_ | 1866915234769272832 |
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| author | Bousquet-Mélou, Mireille Courtiel, Julien |
| author_facet | Bousquet-Mélou, Mireille Courtiel, Julien |
| contents | We address the enumeration of p-valent planar maps equipped with a spanning forest, with a weight z per face and a weight u per connected component of the forest. Equivalently, we count p-valent maps equipped with a spanning tree, with a weight z per face and a weight μ:=u+1 per internally active edge, in the sense of Tutte; or the (dual) p-angulations equipped with a recurrent sandpile configuration, with a weight z per vertex and a variable μ:=u+1 that keeps track of the level of the configuration. This enumeration problem also corresponds to the limit q -> 0 of the q-state Potts model on p-angulations. Our approach is purely combinatorial. The associated generating function, denoted F(z,u), is expressed in terms of a pair of series defined implicitly by a system involving doubly hypergeometric series. We derive from this system that F(z,u) is differentially algebraic in z, that is, satisfies a differential equation in z with polynomial coefficients in z and u. This has recently been proved to hold for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For u >= -1, we study the singularities of F(z,u) and the corresponding asymptotic behaviour of its n-th coefficient. For u>0, we find the standard asymptotic behaviour of planar maps, with a subexponential term in n^{-5/2}. At u=0 we witness a phase transition with a term n^{-3}. When u\in[-1,0), we obtain an extremely unusual behaviour in n^{-3}(\ln n)^{-2}. To our knowledge, this is a new "universality class" for planar maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1306_4536 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Spanning forests in regular planar maps Bousquet-Mélou, Mireille Courtiel, Julien Combinatorics We address the enumeration of p-valent planar maps equipped with a spanning forest, with a weight z per face and a weight u per connected component of the forest. Equivalently, we count p-valent maps equipped with a spanning tree, with a weight z per face and a weight μ:=u+1 per internally active edge, in the sense of Tutte; or the (dual) p-angulations equipped with a recurrent sandpile configuration, with a weight z per vertex and a variable μ:=u+1 that keeps track of the level of the configuration. This enumeration problem also corresponds to the limit q -> 0 of the q-state Potts model on p-angulations. Our approach is purely combinatorial. The associated generating function, denoted F(z,u), is expressed in terms of a pair of series defined implicitly by a system involving doubly hypergeometric series. We derive from this system that F(z,u) is differentially algebraic in z, that is, satisfies a differential equation in z with polynomial coefficients in z and u. This has recently been proved to hold for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For u >= -1, we study the singularities of F(z,u) and the corresponding asymptotic behaviour of its n-th coefficient. For u>0, we find the standard asymptotic behaviour of planar maps, with a subexponential term in n^{-5/2}. At u=0 we witness a phase transition with a term n^{-3}. When u\in[-1,0), we obtain an extremely unusual behaviour in n^{-3}(\ln n)^{-2}. To our knowledge, this is a new "universality class" for planar maps. |
| title | Spanning forests in regular planar maps |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1306.4536 |