Critical exponents on Fortuin--Kasteleyn weighted planar maps

TL;DR

This paper rigorously determines critical exponents for cluster interfaces and enclosed areas in random planar maps weighted by the Fortuin--Kasteleyn model, linking these exponents to SLE parameters and confirming theoretical predictions.

Contribution

It provides the first rigorous calculation of critical exponents for FK-weighted planar maps, connecting them to SLE theory and KPZ relations.

Findings

Critical exponent for cluster interface length: rac{4}{ heta} arccos( rac{ ext{sqrt}(2 - ext{sqrt}(q))}{2} )

Exponent for enclosed area: 1 for all q in (0,4)

Consistency with SLE curve dimensions and duality via KPZ formula

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 $$ \frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{\kappa'}{8}. $$ where $\kappa' $ 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.