Spanning forests in regular planar maps

TL;DR

This work develops a purely combinatorial framework to enumerate $p$-valent planar maps endowed with a spanning forest, captured by the generating function $F(z,u)$ that tracks faces and forest components. By deriving an implicit system involving doubly hypergeometric series and introducing auxiliary series $R$ and $S$, the authors prove that $F(z,u)$ is differentially algebraic in $z$ and obtain explicit differential equations in the important cases $p=4$ and $p=3$, along with a comprehensive singularity-analysis-based account of asymptotics. The asymptotic behavior undergoes a phase transition at $u=0$ and reveals a novel universality class for $u\in[-1,0)$, where $F(z,u)$ is not D-finite, contrasting with the standard planar-map growth for $u>0$. The paper further connects forested maps to Tutte polynomials, sandpile configurations, and Potts-model limits ($q\to 0$), and extends to large random-map statistics, including component counts and internally active edges, thereby enriching the understanding of random planar structures with embedded forest-like constraints. The methods blend rich combinatorial decompositions (forests in contracted maps, enriched blossoming trees) with analytic tools (logarithmic and implicit-function inversions and singularity analysis), yielding precise growth rates and unveiling new universality phenomena in map enumeration.

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 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 n-th 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\in[-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.

Figures (14)

• Figure 1: (a) A rooted planar map and its dual (rooted at the dual corner). (b) A 4-valent leaf-rooted tree.
• Figure 2: The 10 forested cubic maps with 3 faces.
• Figure 3: The edges of a tree-rooted map are naturally order by walking around the tree. The active edges are those labelled 1, 3, 6 and 9.
• Figure 4: (a) A 4-valent forested map with 9 faces and 2 non-root components. (b) The same map, after contraction of the forest. (c) Assembling the 3 trees gives the original forested map.
• Figure 5: A quasi-cubic forested map with 6 faces and 4 non-root components.

Theorems & Definitions (62)

• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm:equations']}
• Proposition 3.3
• Lemma 3.4
• proof
• proof : Proof of Proposition \ref{['prop:G']}
• Proposition 3.5
• Lemma 3.6