The Annals of Probability

Extreme nesting in the conformal loop ensemble

Jason Miller, Samuel S. Watson, and David B. Wilson

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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.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 1013-1052.

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

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Zentralblatt MATH identifier

Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60F10: Large deviations
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 37A25: Ergodicity, mixing, rates of mixing

SLE CLE conformal loop ensemble Gaussian free field


Miller, Jason; Watson, Samuel S.; Wilson, David B. Extreme nesting in the conformal loop ensemble. Ann. Probab. 44 (2016), no. 2, 1013--1052. doi:10.1214/14-AOP995.

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