Spanning forests in regular planar maps (conference version)

Bousquet-Mélou, Mireille ; Courtiel, Julien

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) (2013).

Résumé

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 component of the forest. Equivalently, we count regular 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. This enumeration problem corresponds to the limit $q \rightarrow 0$ of the $q$-state Potts model on the (dual) $p$-angulations. Our approach is purely combinatorial. The generating function, denoted by $F(z,u)$, is expressed in terms of a pair of series defined by an implicit system involving doubly hypergeometric functions. We derive from this system that $F(z,u)$ is $\textit{differentially algebraic}$, that is, satisfies a differential equation (in $z$) with polynomial coefficients in $z$ and $u$. This has recently been proved for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For $u \geq -1$, we study the singularities of $F(z,u)$ and the corresponding asymptotic behaviour of its $n^{\mathrm{th}}$ coefficient. For $u > 0$, we find the standard asymptotic behaviour of planar maps, with a subexponential factor $n^{-5/2}$. At $u=0$ we witness a phase transition with a factor $n^{-3}$. When $u \in[-1,0)$, we obtain an extremely unusual behaviour in $n^{-3}/(\log n)^2$. To our knowledge, this is a new ''universality class'' of planar maps.

Détail

DOI : 10.46298/dmtcs.12808