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.
Reviewer: Achim Klenke (Mainz)
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 |
Keywords:
conformal loop ensemblesReferences:
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