Decomposition of Brownian loop-soup clusters

Understand

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.