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

Journal of Theoretical Probability volume 33, pages 828–865 (2020)Cite this article

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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References

  1. Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory, vol. 371. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  2. Benoist, S., Hongler, C.: The Scaling Limit of Critical Ising Interfaces is CLE(3). arXiv:1604.06975 (2016)

  3. Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for \(q\ge 1\). Probab. Theory Rel. Fields 153(3–4), 511–542 (2012)

    MATH  Google Scholar 

  4. Bertoin, J.: Lévy Processes, vol. 121. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  5. Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A., Smirnov, S.: Convergence of Ising interfaces to Schramm’s SLE curves. Compt. R. Math. 352(2), 157–161 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Chelkak, D., Duminil-Copin, H., Hongler, C.: Crossing probabilities in topological rectangles for the critical planar FK-Ising model. Electron. J. Probab. 21(5), 28 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Chelkak, D.: Robust discrete complex analysis: a toolbox. Ann. Probab. 44(1), 628–683 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Chelkak, D.: 2D Ising model: correlation functions at criticality via Riemann-type boundary value problems. In: European Congress of Mathematics, Berlin, 18–22 July 2016, pp. 235–256. European Mathematical Society, Zurich (2018)

  9. Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. (2) 181(3), 1087–1138 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Camia, F., Newman, C.M.: Two-dimensional critical percolation: the full scaling limit. Commun. Math. Phys. 268(1), 1–38 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Camia, F., Newman, C.M.: Critical percolation exploration path and \({\rm SLE}_6\): a proof of convergence. Probab. Theory Rel. Fields 139(3–4), 473–519 (2007)

    MATH  Google Scholar 

  12. Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Duminil-Copin, H.: Parafermionic observables and their applications to planar statistical physics models. Ensaios Matematicos 25, 1–371 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Continuity of the phase transition for planar random-cluster and Potts models with \(1 \le q \le 4\). Commun. Math. Phys. 349(1), 47–107 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Garban, C., Wu, H.: Dust Analysis in FK-Ising Percolation and Convergence to SLE (16/3, 16/3–6) (Unpublished manuscript). http://math.univ-lyon1.fr/~garban/Fichiers/FKIsing_onearm_manuscript.pdf

  16. Goggin, E.M.: Convergence in distribution of conditional expectations. Ann. Probab. 22(2), 1097–1114 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Kemppainen, A., Smirnov, S.: Conformal invariance of boundary touching loops of FK Ising model. arXiv:1509.08858 (2015)

  18. Kemppainen, A., Smirnov, S.: Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. arXiv:1609.08527 (2016)

  19. Kemppainen, A., Smirnov, S.: Random curves, scaling limits and Loewner evolutions. Ann. Probab. 45(2), 698–779 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  21. Lawler, G.F., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electron. J. Probab. 7(2), 13 (2002). (electronic)

    MathSciNet  MATH  Google Scholar 

  22. Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Lupu, T.: From loop clusters and random interlacements to the free field. Ann. Probab. 44(3), 2117–2146 (2016)

    MathSciNet  MATH  Google Scholar 

  24. McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)

    MATH  Google Scholar 

  25. Miller, J., Sheffield, S.: Imaginary geometry I: interacting SLEs. Probab. Theory Rel. Fields 164(3–4), 553–705 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Miller, J., Sheffield, S.: Imaginary geometry II: reversibility of \(\operatorname{SLE}_\kappa (\rho _1;\rho _2)\) for \(\kappa \in (0,4)\). Ann. Probab. 44(3), 1647–1722 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Newman, M.H.A.: Elements of the Topology of Plane Sets of Points, 2nd edn. Dover Publications Inc, New York (1992)

    MATH  Google Scholar 

  28. Pommerenke, C.: Boundary behaviour of conformal maps, vol. 299. Grundlehren der Mathematischen Wissenschaften Springer-Verlag, Berlin (1992)

    MATH  Google Scholar 

  29. Revuz, D., Yor, M.: Continuous martingales and Brownian motion, vol. 293, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin (1994)

    MATH  Google Scholar 

  30. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202(1), 21–137 (2009)

    MathSciNet  MATH  Google Scholar 

  32. Schramm, O., Sheffield, S., Wilson, D.B.: Conformal radii for conformal loop ensembles. Commun. Math. Phys. 288(1), 43–53 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Schramm, O., Wilson, D.B.: SLE coordinate changes. N. Y. J. Math. 11, 659–669 (2005). (electronic)

    MathSciNet  MATH  Google Scholar 

  34. Sheffield, S.: Exploration trees and conformal loop ensembles. Duke Math. J. 147(1), 79–129 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)

    MathSciNet  MATH  Google Scholar 

  36. Smirnov, S.: Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. Math. 172, 1435–1467 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Smirnov, S., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8(5–6), 729–744 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Werner, W.: Lectures on two-dimensional critical percolation (2007)

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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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  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, Lyon, France

    Christophe Garban

  2. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Hao Wu

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  1. Christophe Garban
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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