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SLE as a Mating of Trees in Euclidean Geometry

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Abstract

The mating of trees approach to Schramm–Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier et al. (Liouville quantum gravity as a mating of trees, 2014. arXiv:1409.7055). In this paper we consider the mating of trees approach to SLE in Euclidean geometry. Let \({\eta}\) be a whole-plane space-filling SLE with parameter \({\kappa > 4}\), parameterized by Lebesgue measure. The main observable in the mating of trees approach is the contour function, a two-dimensional continuous process describing the evolution of the Minkowski content of the left and right frontier of \({\eta}\). We prove regularity properties of the contour function and show that (as in the LQG case) it encodes all the information about the curve \({\eta}\). We also prove that the uniform spanning tree on \({\mathbb{Z}^2}\) converges to SLE8 in the natural topology associated with the mating of trees approach.

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References

  1. Abraham R., Delmas J.-F., Hoscheit P.: A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18(14), 21 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Aldous D.: The continuum random tree. I. Ann. Probab. 19(1), 1–28 (1991)

    Article  MathSciNet  Google Scholar 

  3. Aldous, D.: The continuum random tree. II. An overview. In: Stochastic Analysis (Durham, 1990), Volume 167 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, pp. 23–70 (1991)

  4. Aldous D.: The continuum random tree. III. Ann. Probab. 21(1), 248–289 (1993)

    Article  MathSciNet  Google Scholar 

  5. Barlow M.T., Croydon D.A., Kumagai T.: Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree. Ann. Probab. 45(1), 4–55 (2017)

    Article  MathSciNet  Google Scholar 

  6. Benoist, S.: Natural parametrization of SLE: the Gaussian free field point of view. ArXiv e-prints, August 2017, arXiv:1708.03801

  7. Bernardi, O.: Bijective counting of tree-rooted maps and shuffles of parenthesis systems. Electron. J. Comb., 14(1): Research Paper 9, 36 pp. (electronic), (2007).

  8. Berestycki N.: An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22(27), 12 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Barlow M.T., Masson R.: Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab. 38(6), 2379–2417 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  12. David F., Kupiainen A., Rhodes R., Vargas V.: Liouville quantum gravity on the Riemann sphere. Commun. Math. Phys. 342(3), 869–907 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  13. Duplantier, B., Miller, J., Sheffield, S.: Liouville quantum gravity as a mating of trees. ArXiv e-prints, September 2014, arXiv:1409.7055

  14. Duplantier B., Sheffield S.: Liouville quantum gravity and KPZ. Invent. Math. 185(2), 333–393 (2011) arXiv:1206.0212

    Article  ADS  MathSciNet  Google Scholar 

  15. Dubédat,J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc. 22(4), 995–1054 (2009). arXiv:0712.3018

    Article  ADS  MathSciNet  Google Scholar 

  16. Durrett, R.: Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 4th edn. Cambridge University Press, Cambridge (2010)

  17. Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)

    Google Scholar 

  18. Gwynne, E., Holden, N., Miller, J.: An almost sure KPZ relation for SLE and Brownian motion. ArXiv e-prints, December 2015, arXiv:1512.01223

  19. Gwynne, E., Holden, N., Miller, J., Sun, X.: Brownian motion correlation in the peanosphere for \({\kappa > 8}\). Ann. Inst. H. Poincaré Probab. Statist. (2016). arXiv:1510.04687 (To appear)

  20. Gwynne, E., Holden, N., Sun, X.: A distance exponent for Liouville quantum gravity. ArXiv e-prints, June 2016, arXiv:1606.01214.

  21. Gwynne, E., Holden, N., Sun, X.: Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense. ArXiv e-prints, March 2016, arXiv:1603.01194

  22. Gwynne, E., Kassel, A., Miller, J., Wilson, D.B.: Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for \({\kappa \geq 8}\). ArXiv e-prints, March 2016, arXiv:1603.09722

  23. Gwynne, E., Mao, C., Sun, X.: Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times. ArXiv e-prints, February 2015, arXiv:1502.00546

  24. Gwynne, E., Sun, X.: Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case. ArXiv e-prints, October 2015, arXiv:1510.06346

  25. Gwynne E., Sun X.: Scaling limits for the critical Fortuin–Kastelyn model on a random planar map II: local estimates and empty reduced word exponent. Electron. J. Probab. 22(45), 55 (2017)

    MATH  Google Scholar 

  26. Kenyon, R., Miller, J., Sheffield, S., Wilson, D.B.: Bipolar orientations on planar maps and SLE12. ArXiv e-prints, November 2015, arXiv:1511.04068

  27. Kenyon R., Miller J., Sheffield S., Wilson D.B.: Six-vertex model and Schramm–Loewner evolution. Phys. Rev. E 95, 052146 (2017)

    Article  ADS  Google Scholar 

  28. Kemppainen, A., Smirnov, S.: Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. ArXiv e-prints, September 2016, arXiv:1609.08527

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

  30. Le Gall J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)

    Article  MathSciNet  Google Scholar 

  31. Lawler G.F., Rezaei M.A.: Minkowski content and natural parameterization for the Schramm–Loewner evolution. Ann. Probab. 43(3), 1082–1120 (2015)

    Article  MathSciNet  Google Scholar 

  32. Lawler G.F., Sheffield S.: A natural parametrization for the Schramm–Loewner evolution. Ann. Probab. 39(5), 1896–1937 (2011) arXiv:0906.3804

    Article  MathSciNet  Google Scholar 

  33. 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) arXiv:math/0112234

    Article  MathSciNet  Google Scholar 

  34. Li, Y., Sun, X., Watson, S.S.: Schnyder woods, SLE(16), and Liouville quantum gravity. ArXiv e-prints, May 2017, arXiv:1705.03573

  35. Lawler, G.F., Viklund, F.: Convergence of loop-erased random walk in the natural parametrization. ArXiv e-prints, March 2016, arXiv:1603.05203

  36. Lawler, G.F., Viklund, F.: Convergence of radial loop-erased random walk in the natural parametrization. ArXiv e-prints, March 2017, arXiv:1703.03729

  37. Masson R.: The growth exponent for planar loop-erased random walk. Electron. J. Probab. 14(36), 1012–1073 (2009)

    Article  MathSciNet  Google Scholar 

  38. Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. ArXiv e-prints, May 2016, arXiv:1605.03563

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

    Article  MathSciNet  Google Scholar 

  40. Miller J., Sheffield S.: Imaginary geometry II: reversibility of \({{\rm SLE}_\kappa(\rho_1;\rho_2)}\) for \({\kappa\in(0,4)}\). Ann. Probab. 44(3), 1647–1722 (2016)

    Article  MathSciNet  Google Scholar 

  41. Miller J., Sheffield S.: Imaginary geometry III: reversibility of \({{\rm SLE}_\kappa}\) for \({\kappa\in(4,8)}\). Ann. Math. (2) 184(2), 455–486 (2016)

    Article  MathSciNet  Google Scholar 

  42. Miller J., Sheffield S.: Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Relat. Fields 169(3–4), 729–869 (2017)

    Article  MathSciNet  Google Scholar 

  43. Mullin R.C.: On the enumeration of tree-rooted maps. Can. J. Math. 19, 174–183 (1967)

    Article  MathSciNet  Google Scholar 

  44. Mackey, B., Zhan, D.: Multipoint Estimates for Radial and Whole-plane SLE. ArXiv e-prints, October 2017, arXiv:1710.04261.

  45. Prohorov Y.V.: Convergence of random processes and limit theorems in probability theory. Teor. Veroyatnost. i Primenen. 1, 177–238 (1956)

    MathSciNet  Google Scholar 

  46. Rhodes R., Vargas V.: Gaussian multiplicative chaos and applications: a review. Probab. Surv. 11, 315–392 (2014) arXiv:1305.6221

    Article  MathSciNet  Google Scholar 

  47. Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000) arXiv:math/9904022

    Article  MathSciNet  Google Scholar 

  48. Sheffield S.: Gaussian free fields for mathematicians. Probab. Theory Related Fields 139(3-4), 521–541 (2007) arXiv:math/0312099

    Article  MathSciNet  Google Scholar 

  49. Sheffield S.: Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab. 44(5), 3474–3545 (2016)

    Article  MathSciNet  Google Scholar 

  50. Sheffield S.: Quantum gravity and inventory accumulation. Ann. Probab. 44(6), 3804–3848 (2016)

    Article  MathSciNet  Google Scholar 

  51. 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) arXiv:0909.4499

    Article  ADS  MathSciNet  Google Scholar 

  52. Schramm O., Sheffield S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202(1), 21–137 (2009) arXiv:math/0605337

    Article  MathSciNet  Google Scholar 

  53. Schramm O., Sheffield S.: A contour line of the continuum Gaussian free field. Probab. Theory Related Fields 157(1-2), 47–80 (2013) arXiv:math/0605337

    Article  MathSciNet  Google Scholar 

  54. Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996). ACM, New York, pp. 296–303 (1996)

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Nina Holden

  2. Columbia University, New York, NY, USA

    Xin Sun

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  1. Nina Holden
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  2. Xin Sun
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Correspondence to Xin Sun.

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Communicated by H. Duminil-Copin

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Holden, N., Sun, X. SLE as a Mating of Trees in Euclidean Geometry. Commun. Math. Phys. 364, 171–201 (2018). https://doi.org/10.1007/s00220-018-3149-1

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