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 ∂ (which is known to be an SLE₄-type loop), then the union of all excursions away from ∂ by all the Brownian loops in the loop-soup that touch ∂ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by ∂. 

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.