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

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal SLE_κ ( κ -6) with κ =16/3 . Our proof follows the classical excursion construction of SLE_κ (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 that of Kemppainen and Smirnov (Conformal invariance of boundary touching loops of FK–Ising model. arXiv:1509.08858 , 2015; Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. arXiv:1609.08527 , 2016) as it only relies on the convergence to the chordal SLE_κ process in Dobrushin boundary condition and does not require the introduction of a new observable. Still, it relies crucially on several ingredients: the powerful topological framework developed in Kemppainen and Smirnov (Ann Probab 45(2):698–779, 2017) as well as its follow-up paper Chelkak et al. (Compt R Math 352(2):157–161, 2014), the strong RSW Theorem from Chelkak et al. (Electron. J. Probab. 21(5):28, 2016), the proof is inspired from the appendix A in Benoist and Hongler (The scaling limit of critical Ising interfaces is CLE(3). arXiv:1604.06975 , 2016). One important emphasis of this paper is to carefully write down some properties which are often considered folklore in the literature but which are only justified so far by hand-waving arguments. The main examples of these are: the convergence of natural discrete stopping times to their continuous analogues. (The usual hand-waving argument destroys the spatial Markov property.) the fact that the discrete spatial Markov property is preserved in the scaling limit. (The enemy being that 𝔼 [X_n | Y_n ] does not necessarily converge to 𝔼 [X | Y ] when (X_n,Y_n)→ (X,Y) .) We end the paper with a detailed sketch of the convergence to radial SLE_κ ( κ -6) when κ =16/3 as well as the derivation of Onsager’s one-arm exponent 1 / 8.