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: If one conditions a loop-soup cluster on its outer boundary partial derivative (which is known to be an SLE4-type loop), then the union of all excursions away from partial derivative by all the Brownian loops in the loop-soup that touch partial derivative is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by partial derivative. A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with CLE4 via level lines (by Miller-Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the CLE4 with loop-soups via loop-soup clusters (by Sheffield and Werner) can be made to coincide. An instrumental role in our proof of this fact is played by Lupu's description of CLE4 as limits of discrete loop-soup clusters.