Résumé : We consider the 3-state Potts generating function $T(\nu, w)$ of planar triangulations; that is, the series in $\nu$ and $w$ counting planar triangulations with vertices coloured in $3$ colours, weighted by their size and by the number of monochromatic edges (variable $\nu$).

This series was proved to be algebraic $15$ years ago: 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 progresses on the effective solution of DDEs, the exact value of had remained unknown so far. We have determined at last this exact value, proving that $T(\nu, w)$ satisfies a polynomial equation of degree $11$ in $T$. From this we determine the critical value of $\nu$ and the associated exponent.

Another approach, applied to the heavier case of general planar maps (still $3$-coloured) yields an equation of degree $22$.

Joint work with Mireille Bousquet-Mélou (LaBRI, Bordeaux)

Dernière modification : Wednesday 17 September 2025

Contact pour cette page : Cyril.Banderier at lipn.univ-paris13.fr