Three-dimensional Brownian loop soup clusters

TL;DR

This paper investigates the properties of Brownian loop soup clusters in three-dimensional space, revealing a phase transition for unbounded clusters, and compares these clusters with finite-range systems, providing new tools for analysis.

Contribution

It establishes the existence of a phase transition in 3D Brownian loop soup clusters and introduces decomposition formulas and invariance properties for the loop measure.

Findings

Existence of a phase transition for unbounded clusters at large 

Almost sure connectivity into a single cluster for large 

Development of a toolbox for Brownian loop measure in  er 3

Abstract

We study Brownian loop soup clusters in $\mathbb{R}^3$ for an arbitrary intensity $\alpha>0$. 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 $\alpha$ 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.