@article{BOUSQUETMELOU20151,
    Abstract = {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 {$\mu$}:=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 {$\mu$}:=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 nth coefficient. For u>0, we find the standard 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∈[−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. We analyze the phase transition at u=0 in terms of the sandpile model on large maps, and find it to be of infinite order.},
    Author = {Bousquet-M{\'e}lou, Mireille and Courtiel, Julien},
    File = {Spanning forests in regular planar maps - 1-s2.0-S0097316515000503-main - a.pdf},
    ISSN = {0097-3165},
    Journal = {Journal of Combinatorial Theory, Series A},
    Keywords = {Enumeration, Planar maps, Tutte polynomial, Spanning forests},
    Pages = {1-59},
    Title = {Spanning forests in regular planar maps},
    URL = {https://www.sciencedirect.com/science/article/pii/S0097316515000503},
    Volume = {135},
    Year = {2015},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0097316515000503},
    bdsk-url-2 = {https://doi.org/10.1016/j.jcta.2015.04.002},
    date-added = {2023-03-02 09:41:10 +0100},
    date-modified = {2023-03-02 09:41:10 +0100},
    doi = {10.1016/j.jcta.2015.04.002}
}

@article{BOUSQUETMELOU20151, Abstract = {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 {$\mu$}:=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 {$\mu$}:=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 nth coefficient. For u>0, we find the standard 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∈[−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. We analyze the phase transition at u=0 in terms of the sandpile model on large maps, and find it to be of infinite order.}, Author = {Bousquet-M{\'e}lou, Mireille and Courtiel, Julien}, File = {Spanning forests in regular planar maps - 1-s2.0-S0097316515000503-main - a.pdf}, ISSN = {0097-3165}, Journal = {Journal of Combinatorial Theory, Series A}, Keywords = {Enumeration, Planar maps, Tutte polynomial, Spanning forests}, Pages = {1-59}, Title = {Spanning forests in regular planar maps}, URL = {https://www.sciencedirect.com/science/article/pii/S0097316515000503}, Volume = {135}, Year = {2015}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0097316515000503}, bdsk-url-2 = {https://doi.org/10.1016/j.jcta.2015.04.002}, date-added = {2023-03-02 09:41:10 +0100}, date-modified = {2023-03-02 09:41:10 +0100}, doi = {10.1016/j.jcta.2015.04.002} }