The self-dual point of Fortuin--Kasteleyn planar maps is critical

Abstract

We study the Fortuin--Kasteleyn model of planar maps with parameter $q\in (0,4)$ at and away from its self-dual point. 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 paper we establish a dictionary relating quantities of interest in both approaches. This has several consequences. First, we derive an exact expression for the partition function of the fully packed (colour-symmetric) loop-$O(n)$ model on triangulations, as a function of the outer boundary length. This confirms predictions by Gaudin and Kostov. In particular, this model exhibits critical behaviour, in the sense that the partition function exhibits a power-law decay characteristic of the critical regime at this self-dual point. Secondly, we derive precise asymptotics for geometric features of the self-dual Fortuin--Kasteleyn model of planar maps when $0 < q <4$, such as the exact polynomial tail behaviour of the perimeters of clusters and loops. This sharpens previous results of arXiv:1502.00450 and arXiv:1502.00546. Finally, we prove that Fortuin--Kasteleyn maps undergo a \emph{sharp} phase transition at the self-dual point in the sense that, away from the self-dual point, cluster sizes decay exponentially. This mirrors the celebrated result by Beffara and Duminil-Copin for $q\geq 1$ on a fixed lattice and constitutes the first result establishing that the self-dual point of Fortuin--Kasteleyn maps is the critical point. A key step of our proof is to use the above dictionary and the probabilistic results to justify rigorously an ansatz commonly assumed in the analytic combinatorics literature.