scholarly journals Extreme nesting in the conformal loop ensemble

2016 ◽  
Vol 44 (2) ◽  
pp. 1013-1052 ◽  
Author(s):  
Jason Miller ◽  
Samuel S. Watson ◽  
David B. Wilson
Keyword(s):  
Conformal Loop Ensemble

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

scholarly journals The conformal loop ensemble nesting field

2015 ◽  
Vol 163 (3-4) ◽  
pp. 769-801 ◽  
Author(s):  
Jason Miller ◽  
Samuel S. Watson ◽  
David B. Wilson
Keyword(s):  
Conformal Loop Ensemble
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