THERMOGAMAS - The Self-Dual Point of Fortuin-Kasteleyn Planar Maps is Critical (William Da Silva)

Abstract: Fortuin-Kasteleyn (FK) maps form a classical model of planar maps decorated with a percolation-like configuration, depending on a weight q>0. This model is also bijectively equivalent to the fully packed (bicoloured) loop-O(n) model on planar triangulations. These have been traditionally studied using either techniques from analytic combinatorics (based in particular on the gasket decomposition of Borot, Bouttier and Guitter) or probabilistic arguments (based on Sheffield's hamburger-cheeseburger bijection). In this work, we establish a dictionary relating quantities of interest in both approaches, which allows us to derive precise asymptotics for geometric features of the self-dual Fortuin--Kasteleyn planar map model when 0<q<4, such as the exact polynomial tail behaviour of cluster and loop sizes. I will explain how this can be used to establish that Fortuin--Kasteleyn maps undergo a sharp phase transition at the self-dual point, therefore proving that the self-dual point of Fortuin--Kasteleyn maps is the critical point. This talk is based on joint work with Nathanaël Berestycki (University of Vienna).