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

Abstract

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal \(\mathrm {SLE}_\kappa ( \kappa -6)\) with \(\kappa =16/3\). Our proof follows the classical excursion construction of \(\mathrm {SLE}_\kappa (\kappa -6)\) 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 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 \(\mathrm {SLE}_{\kappa }\) process in Dobrushin boundary condition and does not require the introduction of a new observable. Still, it relies crucially on several ingredients:

  1. (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),

  2. (b)

    the strong RSW Theorem from Chelkak et al. (Electron. J. Probab. 21(5):28, 2016),

  3. (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. (1)

    the convergence of natural discrete stopping times to their continuous analogues. (The usual hand-waving argument destroys the spatial Markov property.)

  2. (2)

    the fact that the discrete spatial Markov property is preserved in the scaling limit. (The enemy being that \({{\mathbb {E}}\bigl [X_n \bigm | Y_n\bigr ]}\) does not necessarily converge to \({{\mathbb {E}}\bigl [X\bigm | Y\bigr ]}\) when \((X_n,Y_n)\rightarrow (X,Y)\).)

We end the paper with a detailed sketch of the convergence to radial \(\mathrm {SLE}_\kappa ( \kappa -6)\) when \(\kappa =16/3\) as well as the derivation of Onsager’s one-arm exponent 1 / 8.

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Acknowledgements

We wish to thank Vincent Beffara, Dimtry Chelkak, Antti Kemppainen and Avelio Sepúlveda, and Hugo Vanneuville for useful discussions. We thank Jean-Christophe Mourrat for the proof of Lemma 2.1. This work was carried out during visits of H.W. in Lyon funded by the ERC LiKo 676999. We thank an anonymous referee for helpful comments on a draft of this article.

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Correspondence to Hao Wu.

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C. Garban is supported by the ANR grant Liouville ANR-15-CE40-0013 and the ERC Grant LiKo 676999. H. Wu is supported by the Thousand Talents Plan for Young Professionals.

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Garban, C., Wu, H. On the Convergence of FK–Ising Percolation to \(\mathrm {SLE}(16/3, (16/3)-6)\). J Theor Probab 33, 828–865 (2020). https://doi.org/10.1007/s10959-019-00950-9

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  • DOI: https://doi.org/10.1007/s10959-019-00950-9

Keywords

  • Schramm–Loewner evolution
  • FK percolation
  • Bessel process

Mathematics Subject Classification (2010)

  • 60J67