Three-dimensional Brownian loop soup clusters
Résumé
We study Brownian loop soup clusters in R^3 for an arbitrary intensity α >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 α 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 R^d , d ≥ 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.
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