  The two-dimensional Brownian loop-soup is a Poissonian random collection of
loops in a planar domain with an intensity parameter c. When c is not greater
than 1, we show that the outer boundaries of the loop clusters are disjoint
simple loops (when c>1, there is almost surely only one cluster) that satisfy
certain conformal restriction axioms. We prove various results about
loop-soups, cluster sizes, and the c=1 phase transition.
  Combining this with the results of another paper of ours on the Markovian
characterization of simple conformal loop ensembles (CLE), this proves that
these outer boundaries of clusters of Brownian loops are in fact SLE(k) loops
for k in (8/3, 4]. More generally, it completes the proof of the fact that the
following three descriptions of simple CLEs (proposed in earlier works by the
authors) are equivalent:
  (1) The random loop ensembles traced by branching Schramm-Loewner Evolution
(SLE(k)) curves for k in (8/3, 4].
  (2) The outer-cluster-boundary ensembles of Brownian loop-soups.
  (3) The (only) random loop ensembles satisfying the conformal restriction
axioms.
