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 $\kappa\le4$, 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.