Counting coloured planar maps: differential equations

TL;DR

The paper proves that the generating function counting $q$-coloured planar maps by edges and monochromatic edges is differentially algebraic for an indeterminate $q$, and provides an explicit differential system characterizing this function. It extends the result to planar triangulations and recovers Tutte’s differential equation for properly coloured triangulations as a special case, while also exploring specializations such as four colours, spanning forests, and the self-dual Potts model. The authors develop a framework based on invariant equations with catalytic variables, Chebyshev polynomials, and the construction of $m-2$ auxiliary series $I_i$ to derive a robust differential-system description. They show uniqueness and analyze potential singularities, and they provide detailed treatments of several significant special cases where the differential equations can be reduced to low orders, thereby linking to classical results and physical interpretations. The work advances the understanding of Potts-model enumerations on random planar maps and provides a computational-pathway (via a Maple session) to explicit differential relations across broad parameter regimes.

Abstract

We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial polynomial differential equation withrespect to the edge variable. We give explicitly a differential systemthat characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. Instatistical physics terms, we solvethe q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating functionwas proved to be algebraic for certain values of q,including q=1, 2 and 3. It isknown to be transcendental in general. In contrast, our differential system holds for an indeterminate q.For certain special cases of combinatorial interest (four colours; properq-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.

Figures (3)

• Figure 1: A rooted planar map and its dual (rooted at the dual corner).
• Figure 2: The planar maps with $e=0,1,$ and $2$ edges, and their Potts polynomials (divided by $q$). The maps are shown unrooted, and the numbers in parentheses indicate the number of corresponding rooted maps.
• Figure 3: The rooted near-triangulations of outer degree 1 with $v=2$ and $v=3$ vertices, and their Potts polynomials (divided by $q$).

Theorems & Definitions (29)

• Theorem 1
• Lemma 2
• proof : Proof of Lemma \ref{['lem:initial-Cr']}.
• Lemma 3
• proof : Proof of Lemma \ref{['lem:count']}
• Lemma 4
• proof
• Proposition 5
• proof
• Theorem 6