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.

To appear in J. Europ. Math. Soc

References in corpus (3)

• Conformal radii for conformal loop ensembles
• CLE percolations
• Convergence of the two-dimensional random walk loop soup clusters to CLE

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• Lecture notes on the Gaussian Free Field
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• Covariant Symanzik identities
• Multiplicative chaos of the Brownian loop soup
• The trunks of CLE(4) explorations
• Generalized disconnection exponents
• Liouville quantum gravity weighted by conformal loop ensemble nesting statistics
• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field
• Crossing exponent in the Brownian loop soup
• Dimension of two-valued sets via imaginary chaos
• Cluster explorations of the loop soup on a metric graph related to the Gaussian free field