2018/02/12 by Garban, Christophe, Wu, Hao
#60J67 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1802.03939
We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLEκ( κ-6) with κ=16/3. Our proof follows the classical excursion-construction of SLEκ(κ-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 SLEκ 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 𝔼[Xn | Yn] does not necessarily converge to 𝔼[X| Y] when (Xn,Yn)→ (X,Y)). We end the paper with a detailed sketch of the convergence to radial SLEκ( κ-6) when κ=16/3 as well as the derivation of Onsager's one-arm exponent 1/8.