Decomposition of Brownian loop-soup clusters

  • Wei Qian

    University of Cambridge, UK and ETH Zürich, Switzerland
  • Wendelin Werner

    ETH Zürich, Switzerland
Decomposition of Brownian loop-soup clusters cover
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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 (which is known to be an SLE4_4-type loop), then the union of all excursions away from \partial by all the Brownian loops in the loop-soup that touch \partial is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by \partial.

A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with CLE4_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 CLE4_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 CLE4_4 as limits of discrete loop-soup clusters.

Cite this article

Wei Qian, Wendelin Werner, Decomposition of Brownian loop-soup clusters. J. Eur. Math. Soc. 21 (2019), no. 10, pp. 3225–3253

DOI 10.4171/JEMS/902