Counting coloured planar maps: differential equations
Résumé
We address the enumeration of q-coloured planar maps counted by
the 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 with
respect to the edge variable. We give explicitly a differential system
that characterizes this series. We then
prove a similar result for planar triangulations, thus generalizing
a result of Tutte dealing with their proper q-colourings. In
statistical physics terms, we solve
the q-state Potts model on random planar lattices.
This work follows a first paper by the same authors, where the generating function
was proved to be algebraic for certain values of q,
including q=1, 2 and 3. It is
known to be transcendental in general. In contrast, our
differential system holds for an indeterminate q.
For certain special cases of combinatorial interest (four colours; proper
q-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.
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