The Annals of Probability

Extreme nesting in the conformal loop ensemble

Jason Miller, Samuel S. Watson, and David B. Wilson

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Abstract

The conformal loop ensemble $\operatorname{CLE}_{\kappa}$ with parameter $8/3<\kappa<8$ is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. Given $\kappa$ and $\nu$, we compute the almost-sure Hausdorff dimension of the set of points $z$ for which the number of CLE loops surrounding the disk of radius $\varepsilon $ centered at $z$ has asymptotic growth $\nu\log(1/\varepsilon )$ as $\varepsilon \to0$. By extending these results to a setting in which the loops are given i.i.d. weights, we give a CLE-based treatment of the extremes of the Gaussian free field.

Article information

Source
Ann. Probab. Volume 44, Number 2 (2016), 1013-1052.

Dates
Received: January 2014
Revised: December 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960390

Digital Object Identifier
doi:10.1214/14-AOP995

Mathematical Reviews number (MathSciNet)
MR3474466

Zentralblatt MATH identifier
1347.60061

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60F10: Large deviations
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 37A25: Ergodicity, mixing, rates of mixing

Keywords
SLE CLE conformal loop ensemble Gaussian free field

Citation

Miller, Jason; Watson, Samuel S.; Wilson, David B. Extreme nesting in the conformal loop ensemble. Ann. Probab. 44 (2016), no. 2, 1013--1052. doi:10.1214/14-AOP995. https://projecteuclid.org/euclid.aop/1457960390.


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References

  • [1] Beffara, V. and Duminil-Copin, H. (2012). The self-dual point of the two-dimensional random-cluster model is critical for $q\geq1$. Probab. Theory Related Fields 153 511–542.
  • [2] Bramson, M. and Zeitouni, O. (2012). Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 1–20.
  • [3] Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1–38.
  • [4] Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and $\mathrm{SLE}_{6}$: A proof of convergence. Probab. Theory Related Fields 139 473–519.
  • [5] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2001). Thick points for planar Brownian motion and the Erdős–Taylor conjecture on random walk. Acta Math. 186 239–270.
  • [6] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
  • [7] Falconer, K. (2003). Fractal Geometry, 2nd ed. Wiley, Hoboken, NJ.
  • [8] Fortuin, C. M. and Kasteleyn, P. W. (1972). On the random-cluster model. I. Introduction and relation to other models. Physica 57 536–564.
  • [9] Frostman, O. (1935). Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Math. Sem. 3 1–118.
  • [10] Hu, X., Miller, J. and Peres, Y. (2010). Thick points of the Gaussian free field. Ann. Probab. 38 896–926.
  • [11] Kager, W. and Nienhuis, B. (2004). A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115 1149–1229.
  • [12] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI.
  • [13] Miller, J. and Sheffield, S. (2015). CLE(4) and the Gaussian free field. Preprint.
  • [14] Miller, J., Sun, N. and Wilson, D. B. (2014). The Hausdorff dimension of the CLE gasket. Ann. Probab. 42 1644–1665.
  • [15] Nacu, Ş. and Werner, W. (2011). Random soups, carpets and fractal dimensions. J. Lond. Math. Soc. (2) 83 789–809.
  • [16] Pollard, D. (2007). Minimax theorem. Appendix G to A user’s guide to measure theoretic probability. Available at http://www.stat.yale.edu/~pollard/Courses/602.spring07/MmaxThm.pdf.
  • [17] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883–924.
  • [18] Rhodes, R. and Vargas, V. (2014). Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 315–392.
  • [19] Schramm, O., Sheffield, S. and Wilson, D. B. (2009). Conformal radii for conformal loop ensembles. Comm. Math. Phys. 288 43–53.
  • [20] Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147 79–129.
  • [21] Sheffield, S. and Werner, W. (2012). Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Ann. of Math. (2) 176 1827–1917.
  • [22] Smirnov, S. (2005). Critical percolation and conformal invariance. In XIVth International Congress on Mathematical Physics 99–112. World Sci. Publ., Hackensack, NJ.