Three-dimensional Brownian loop soup clusters

TL;DR

This work analyzes Brownian loop soup clusters in $\mathbb{R}^3$ across all intensities $\alpha>0$, establishing a phase transition between non-percolation and a regime with a unique, unbounded cluster, and showing that in the strong-supercritical phase all loops are connected. Central to the approach is a robust toolbox for Brownian loop measures in $d\ge 3$, including conformal invariance under inversion, loop-measure decompositions, and detailed excursion/bubble measures that enable precise crossing and connectivity estimates. The authors define critical thresholds $\alpha_c,\alpha_u,\alpha_{\rm B2B},\alpha_{\rm tr}$, prove their finiteness and monotonicity, and derive a one-arm exponent $\xi(\alpha)$ governing asymptotics of non-crossing probabilities via a submultiplicative, decoupled framework. They also relate the unbounded percolation phase to local-uniqueness and truncation effects, showing that the supercritical regime exhibits a unique global cluster, with a rigorous treatment of phase-structure through Aizenman–Grimmett-type and Grimmett–Marstrand-type arguments. The results illuminate how continuum percolation in 3D departs from discrete models, provide tools for conformal-invariant analysis in higher dimensions, and establish a foundation for future work on the precise coincidence of multiple phase thresholds.

Abstract

We study Brownian loop soup clusters in $\mathbb{R}^3$ 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 $\mathbb{R}^d$, $d \ge 3$. 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.

Figures (2)

• Figure 6.1: Illustration of the proof of Lemma \ref{['L:finite_critical_point']}
• Figure 9.1: Illustration of some notations used in Section \ref{['SS:seed']}. The large cube represents $Q(r')$. The $x=r'$ face is divided into four squares isometric to $\{r'\} \times [0,r']^2$. In Proposition \ref{['P:seed']}, we start with a seed at the origin, meaning that the cube $Q(r)$ contains a unique very well connected cluster of $\{\wp \in \mathcal{L}_{R',R"}^\alpha: \wp \cap Q(r-R"/2)\neq\varnothing\}$; see Definition \ref{['D:seed']}. Proposition \ref{['P:seed']} then asserts that with high probability, we can find a seed within some $Q(x,r)$ with $x \in \{r'\} \times [0,r']^2$ which is connected to the seed at the origin in $\mathcal{L}_{1,R}^\alpha\cap Q(r'-r)$.

Theorems & Definitions (83)

• Definition 1.1: Clusters
• Theorem 1.2: Phase transition
• Remark 1.3
• Theorem 1.4: One-arm exponent
• Theorem 1.5: Truncated model and local uniqueness
• Conjecture 1.6
• Lemma 2.1: Palm's formula
• Lemma 2.2: FKG inequality
• Lemma 2.3: Square-root trick
• Lemma 3.1