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July 2014 The Hausdorff dimension of the CLE gasket
Jason Miller, Nike Sun, David B. Wilson
Ann. Probab. 42(4): 1644-1665 (July 2014). DOI: 10.1214/12-AOP820

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.

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Jason Miller. Nike Sun. David B. Wilson. "The Hausdorff dimension of the CLE gasket." Ann. Probab. 42 (4) 1644 - 1665, July 2014. https://doi.org/10.1214/12-AOP820

Information

Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1305.60078
MathSciNet: MR3262488
Digital Object Identifier: 10.1214/12-AOP820

Subjects:
Primary: 60J67
Secondary: 60D05

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

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 4 • July 2014
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