Counting Coloured Planar Maps: Differential Equations

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 di erentially algebraic, that is, satis es a non-trivial polynomial di erential equation with respect to the edge variable. We give explicitly a di erential 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 rst 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 di erential 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 di erential equation of order 2 de ning the generating function. For general planar maps, we also obtain a di erential equation of order 3 for the four-colour case and for the self-dual Potts model.