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March 2016 Extreme nesting in the conformal loop ensemble
Jason Miller, Samuel S. Watson, David B. Wilson
Ann. Probab. 44(2): 1013-1052 (March 2016). DOI: 10.1214/14-AOP995


The conformal loop ensemble $\operatorname{CLE}_{\kappa}$ with parameter $8/3<\kappa<8$ is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. Given $\kappa$ and $\nu$, we compute the almost-sure Hausdorff dimension of the set of points $z$ for which the number of CLE loops surrounding the disk of radius $\varepsilon $ centered at $z$ has asymptotic growth $\nu\log(1/\varepsilon )$ as $\varepsilon \to0$. By extending these results to a setting in which the loops are given i.i.d. weights, we give a CLE-based treatment of the extremes of the Gaussian free field.


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Jason Miller. Samuel S. Watson. David B. Wilson. "Extreme nesting in the conformal loop ensemble." Ann. Probab. 44 (2) 1013 - 1052, March 2016.


Received: 1 January 2014; Revised: 1 December 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1347.60061
MathSciNet: MR3474466
Digital Object Identifier: 10.1214/14-AOP995

Primary: 60F10 , 60J67
Secondary: 37A25 , 60D05

Keywords: CLE , Conformal Loop Ensemble , Gaussian free field , SLE

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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