The 3-state Potts model on planar triangulations: Explicit algebraic solution

We consider the 3 -state Potts generating function T(\nu,w) of planar triangulations; that is, the bivariate series that counts planar triangulations with vertices coloured in 3 colours, weighted by their size (number of vertices, recorded by the variable w ) and by the number of monochromatic edges (variable \nu ). This series was proved to be algebraic 15 years ago by Bernardi and the first author: this follows from its link with the solution of a discrete differential equation (DDE) and from general algebraicity results on such equations. However, despite recent progress on the effective solution of DDEs, the exact value of T(\nu,w) has remained unknown so far—except in the case \nu=0 , corresponding to proper colorings, and was solved by Tutte in the sixties. We determine here this exact value, proving that T(\nu,w) satisfies a polynomial equation of degree 11 in T and genus 1 in w and T . We prove that the critical value of \nu is \nu_{c}=1+3/\sqrt{47} , with a critical exponent 6/5 in the series T(\nu_{c}, \cdot) , while the other values of \nu yield the usual map exponent 3/2 . By duality of the planar Potts model, our results also characterize the 3 -state Potts generating function of planar cubic maps, in which all vertices have degree 3 . In particular, the annihilating polynomial, still of degree 11 , that we obtain for properly 3-coloured cubic maps proves a conjecture by Bruno Salvy from 2009.