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SLEs as boundaries of clusters of Brownian loops

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 simple Poissonian percolation clusters: Recall that the Brownian loop-soup (introduced in the paper arxiv:math.PR/0304419 with Greg Lawler) with intensity c defines a Poissonian collection of (simple if one focuses only on the outer boundary) loops in a domain. This random family of (possibly intersecting) loops is conformally invariant (and there are almost surely infinitely many small loops in any sample). We show that there exists a critical value a in (0,1] such that if one colors all the interiors of the loops, the obtained clusters are bounded when c a, one single cluster fills the domain. We prove that for small c, the outer boundaries of the clusters are SLE-type curves where $κ\le 4$ and $c$ related by the usual relation $c=(3κ-8)(6-κ)/2κ$ (i.e. c corresponds to the central charge of the model). Conjecturally, the critical value a is equal to one and corresponds to SLE4 loops, so that this should give for any c in (0,1] a construction of a natural countable family of random disjoint SLE$_κ$ loops (i.e. $κ$ should span $(8/3,4]$), that behaves ``nicely'' under perturbation of the domain. A precise relation between chordal SLE and the loop-soup goes as follows: Consider the sample of a certain restriction measure (i.e. a certain union of Brownian excursions) in a domain, attach to it all the above-described clusters that it intersects. The outer boundary of the obtained set is exactly an SLE$_κ$, if the restriction measure exponent is equal to the highest-weight of the corresponding representation with central charge c.

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Wendelin Werner. 2003-08-18. SLEs as boundaries of clusters of Brownian loops. https://arxiv.org/abs/math/0308164

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