The Hausdorff dimension of the CLE gasket

The conformal loop ensemble CLE. is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter . 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.(8..)(3..8)/(32.) when 4<.<8. Together with the work of Schramm.Sheffield.Wilson [Comm. Math. Phys. 288 (2009) 43.53] giving the upper bound for all . and the work of Nacu.Werner [J. Lond. Math. Soc. (2) 83 (2011) 789.809] giving the matching lower bound for ..4, this completes the determination of the CLE. gasket dimension for all values of . 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.