The Annals of Probability

The Hausdorff dimension of the CLE gasket

Jason Miller, Nike Sun, and David B. Wilson

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Abstract

The conformal loop ensemble $\mathrm{CLE}_{\kappa}$ is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter $\kappa$ varies between $8/3$ and $8$; $\mathrm{CLE}_{8/3}$ is empty while $\mathrm{CLE}_{8}$ is a single space-filling loop. In this work, we study the geometry of the $\mathrm{CLE}$ gasket, the set of points not surrounded by any loop of the $\mathrm{CLE}$. We show that the almost sure Hausdorff dimension of the gasket is bounded from below by $2-(8-\kappa)(3\kappa-8)/(32\kappa)$ when $4<\kappa<8$. Together with the work of Schramm–Sheffield–Wilson [Comm. Math. Phys. 288 (2009) 43–53] giving the upper bound for all $\kappa$ and the work of Nacu–Werner [J. Lond. Math. Soc. (2) 83 (2011) 789–809] giving the matching lower bound for $\kappa\le4$, this completes the determination of the $\mathrm{CLE}_{\kappa}$ gasket dimension for all values of $\kappa$ for which it is defined. The dimension agrees with the prediction of Duplantier–Saleur [Phys. Rev. Lett. 63 (1989) 2536–2537] for the FK gasket.

Article information

Source
Ann. Probab. Volume 42, Number 4 (2014), 1644-1665.

Dates
First available in Project Euclid: 3 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1404394074

Digital Object Identifier
doi:10.1214/12-AOP820

Mathematical Reviews number (MathSciNet)
MR3262488

Zentralblatt MATH identifier
1305.60078

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Schramm–Loewner evolution (SLE) conformal loop ensemble (CLE) gasket

Citation

Miller, Jason; Sun, Nike; Wilson, David B. The Hausdorff dimension of the CLE gasket. Ann. Probab. 42 (2014), no. 4, 1644--1665. doi:10.1214/12-AOP820. https://projecteuclid.org/euclid.aop/1404394074.


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