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

Abstract

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.