SLEs as boundaries of clusters of Brownian loops

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  • WWWendelin Werner
    Wendelin Werner
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Abstract
In this research announcement, 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 SLEκ-type curves where κ∈(8/3,4] and c related by the usual relation c=(3κ−8)(6−κ)/2κ (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 SLEκ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory.

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