On the convergence of FK-Ising Percolation to SLE$(16/3, 16/3-6)$

Abstract

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLE$_\kappa( \kappa-6)$ with $\kappa=16/3$. Our proof follows the classical excursion-construction of SLE$_\kappa(\kappa-6)$ processes in the continuum and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from [KS15, KS16] as it only relies on the convergence to the chordal SLE$_{\kappa}$ process in Dobrushin boundary condition and does not require the introduction of a new observable. Still, it relies crucially on several ingredients: a) the powerful topological framework developed in [KS17] as well as its follow-up paper [CDCH$^+$14], b) the strong RSW Theorem from [CDCH16], c) the proof is inspired from the appendix A in [BH16]. One important emphasis of this paper is to carefully write down some properties which are often considered {\em folklore} in the literature but which are only justified so far by hand-waving arguments. The main examples of these are: 1) the convergence of natural discrete stopping times to their continuous analogues. (The usual hand-waving argument destroys the spatial Markov property). 2) the fact that the discrete spatial Markov property is preserved in the the scaling limit. (The enemy being that $\mathbb{E}[X_n |\, Y_n]$ does not necessarily converge to $\mathbb{E}[X|\, Y]$ when $(X_n,Y_n)\to (X,Y)$). We end the paper with a detailed sketch of the convergence to radial SLE$_\kappa( \kappa-6)$ when $\kappa=16/3$ as well as the derivation of Onsager's one-arm exponent $1/8$.