The Hausdorff dimension of the CLE gasket

The conformal loop ensemble $\mathrm{CLE}_κ$ 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 $8$; $\mathrm{CLE}_{8/3}$ 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 $2-(8-κ)(3κ-8)/(32κ)$ 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 $κ\le4$, this completes the determination of the $\mathrm{CLE}_κ$ 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.

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