Spanning forests in regular planar maps (conference version)

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 μ := u+ 1 per internally active edge, in the sense of Tutte. This enumeration problem corresponds to the limit q → 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 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 ≥ −1, we study the singularities of F (z, u) and the corresponding asymptotic behaviour of its n coefficient. For u > 0, we find the standard asymptotic behaviour of planar maps, with a subexponential factor n. At u = 0 we witness a phase transition with a factor n. When u ∈ [−1, 0), we obtain an extremely unusual behaviour in n/(log n). To our knowledge, this is a new “universality class” of planar maps.