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SLEs as boundaries of clusters of Brownian loops. (English. Abridged French version) Zbl 1029.60085
Summary: We show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities \(c\) of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLE\(_{\kappa}\)-type curves where \(\kappa \in (8/3,4]\) and \(c\) related by the usual relation \(c=(3{\kappa}-8)(6-{\kappa})/2{\kappa}\) (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLE\(_{\kappa}\) loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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