Open Access
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 CLEκ is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter κ varies between 8/3 and 8; CLE8/3 is empty while CLE8 is a single space-filling loop. In this work, we study the geometry of the CLE gasket, the set of points not surrounded by any loop of the CLE. We show that the almost sure Hausdorff dimension of the gasket is bounded from below by 2(8κ)(3κ8)/(32κ) when 4<κ<8. Together with the work of Schramm–Sheffield–Wilson [Comm. Math. Phys. 288 (2009) 43–53] giving the upper bound for all κ and the work of Nacu–Werner [J. Lond. Math. Soc. (2) 83 (2011) 789–809] giving the matching lower bound for κ4, this completes the determination of the CLEκ gasket dimension for all values of κ 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.

Citation

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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

Vol.42 • No. 4 • July 2014
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