The Hausdorff dimension of the CLE gasket

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 $\kappa \le 4$, 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.

Published in at http://dx.doi.org/10.1214/12-AOP820 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• The bulk one-arm exponent for the CLE${}_{{\kappa }^{\prime }}$ percolations