Séminaires : Séminaire Combinatoire, Optimisation, et Interactions

The purpose of this seminar is to foster exchanges within the CO team of IMJ-PRG, and also with the surrounding scientific community. As such, its range of topics should be quite broad. We initially plan one or two sessions per month.

Ce séminaire a pour but de développer les échanges au sein de l'équipe Combinatoire et Optimisation, et avec la communauté scientifique environnante. Ses thèmes seront donc larges. Nous prévoyons un rythme initial de une à deux séances par mois.

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)