Conformal Radii for Conformal Loop Ensembles

The conformal loop ensembles CLE _κ , defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compute the expectation dimension of the CLE _κ gasket, which consists of points not surrounded by any loop, to be , which agrees with the fractal dimension given by Duplantier for the O(n) model gasket. Partially supported by NSF grant DMS0403182.