Three-dimensional Brownian loop soup clusters

Abstract

We study Brownian loop soup clusters in ${\mathbb{R}}^3$ for an arbitrary intensity $\hat{I}\pm >0$. We show the existence of a phase transition for the presence of unbounded clusters and study its basic properties. In particular, we show that, when $\hat{I}\pm$ is sufficiently large, almost surely all the loops are connected into a single cluster. Such a phenomenon is not observed in discrete percolation-type models. In addition, we prove the existence of a one-arm exponent and compare the clusters with the finite-range system obtained by imposing lower and upper bounds on the diameter of the loops. Finally, we provide a toolbox concerning the Brownian loop measure in ${\mathbb{R}}^d$, $d\ge 3$. In particular, we derive decomposition formulas by rerooting the loops in specific ways and show that the loop measure is conformally invariant, generalising results of [Lup18] in dimension 1 and [LW04] in dimension 2.

38 pages, 2 figures. Minor update