Counting coloured planar maps: differential equations

TL;DR

This paper proves that the generating function for counting q-coloured planar maps and triangulations is differentially algebraic, satisfying polynomial differential equations, thus advancing the understanding of their combinatorial and physical properties.

Contribution

It establishes that the generating functions are differentially algebraic for all q, generalizing previous algebraic results and providing explicit differential systems for various cases.

Findings

Generated differential systems characterizing the series.

Proved differential algebraicity for all q, including indeterminate cases.

Derived explicit differential equations for specific cases like four colours.

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.