Simple CLE in doubly connected domains
Author(s)Sheffield, Scott Roger; Watson, Samuel Stewart; Wu, Hao
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We restrict attention to CLEκ for which the loops are simple, i.e. κ ∈ (8/3, 4]. In (Ann. of Math. (2) 176 (2012) 1827-1917), simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLE's, along with the corresponding exploration processes.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Institute of Mathematical Statistics
Sheffield, Scott, Samuel S. Watson, and Hao Wu. “Simple CLE in Doubly Connected Domains.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53, no. 2 (May 2017): 594–615.
Author's final manuscript
Schramm Loewner Evolution, Conformal Loop Ensemble, doubly connected domains, explo- ration process