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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.

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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