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 $\kappa \le 4$ 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). 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$_\kappa$
loops (i.e. $\kappa$ 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$_\kappa$, if the restriction measure exponent is equal to the
highest-weight of the corresponding representation with central charge c.