Spanning forests in regular planar maps

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 nth 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 ∈ [−1, 0), we obtain an extremely unusual behaviour in n−3(lnn)−2. To our knowledge, this is a new “universality class” for planar maps.