Decomposition of Brownian loop-soup clusters

Abstract: 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 $\gamma$. A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with CLE(4) via level-lines (by Miller-Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the CLE(4) 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 CLE(4) as limits of discrete loop-soup clusters.