کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4655185 1632937 2015 59 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Spanning forests in regular planar maps
ترجمه فارسی عنوان
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موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات گسسته و ترکیبات
چکیده انگلیسی

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+1μ:=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+1μ:=u+1 that keeps track of the level   of the configuration. This enumeration problem also corresponds to the limit q→0q→0 of the q-state Potts model on p-angulations.Our approach is purely combinatorial. The associated generating function, denoted F(z,u)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)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≥−1u≥−1, we study the singularities of F(z,u)F(z,u) and the corresponding asymptotic behaviour of its n  th coefficient. For u>0u>0, we find the standard behaviour of planar maps, with a subexponential term in n−5/2n−5/2. At u=0u=0 we witness a phase transition with a term n−3n−3. When u∈[−1,0)u∈[−1,0), we obtain an extremely unusual behaviour in n−3(ln⁡n)−2n−3(ln⁡n)−2. To our knowledge, this is a new “universality class” for planar maps. We analyze the phase transition at u=0u=0 in terms of the sandpile model on large maps, and find it to be of infinite order.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Combinatorial Theory, Series A - Volume 135, October 2015, Pages 1–59
نویسندگان
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