Decomposition of Brownian loop-soup clusters

We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $\gamma$ (which is known to be an SLE(4)-type loop), then the union of all excursions away from $\gamma$ by all the Brownian loops in the loop-soup that touch $\gamma$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $\gamma$.