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Conformal radii for conformal loop ensembles. (English) Zbl 1187.82044
From the abstract: The conformal loop ensembles \(\mathrm{CLE}_\kappa\), defined for \(8/3\leq\kappa\leq8\), 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 \(\mathrm{CLE}_\kappa\) gasket, which consists of points not surrounded by any loop, to be \[ 2 -\frac{ (8 - \kappa)(3\kappa - 8)}{32\kappa}, \] which agrees with the fractal dimension given by Duplantier for the \(O(n)\) model gasket.

MSC:
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60K99 Special processes
60J65 Brownian motion
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