On the Convergence of FK–Ising Percolation to $$\mathrm {SLE}(16/3, (16/3)-6)$$SLE(16/3,(16/3)-6)

Christophe Garban () and Hao Wu ()
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Christophe Garban: Université Claude Bernard Lyon 1
Hao Wu: Tsinghua University

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 828-865

Abstract: Abstract We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal $$\mathrm {SLE}_\kappa ( \kappa -6)$$SLEκ(κ-6) with $$\kappa =16/3$$κ=16/3. Our proof follows the classical excursion construction of $$\mathrm {SLE}_\kappa (\kappa -6)$$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 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 $$\mathrm {SLE}_{\kappa }$$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 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),(b)the strong RSW Theorem from Chelkak et al. (Electron. J. Probab. 21(5):28, 2016),(c)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:(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 scaling limit. (The enemy being that $${{\mathbb {E}}\bigl [X_n \bigm | Y_n\bigr ]}$$E[Xn|Yn] does not necessarily converge to $${{\mathbb {E}}\bigl [X\bigm | Y\bigr ]}$$E[X|Y] when $$(X_n,Y_n)\rightarrow (X,Y)$$(Xn,Yn)→(X,Y).) We end the paper with a detailed sketch of the convergence to radial $$\mathrm {SLE}_\kappa ( \kappa -6)$$SLEκ(κ-6) when $$\kappa =16/3$$κ=16/3 as well as the derivation of Onsager’s one-arm exponent 1 / 8.

Keywords: Schramm–Loewner evolution; FK percolation; Bessel process; 60J67 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00950-9

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