Critical exponents on Fortuin--Kasteleyn weighted planar maps

Abstract

In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be $$ \frac{4}π \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{κ'}{8}. $$ where $κ' $ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop which is shown to be 1 for all values of $q \in (0,4)$. Applying the KPZ formula we find that this value is consistent with the dimension of SLE curves and SLE duality.

35 pages. Proofs are revised, simplified and more details are added. An incorrect area exponent in the main theorem corrected