On loop-soup clusters in three dimensions and higher

We study clusters of the Brownian loop soup on $\mathbb{R}^3$ and of random walk and metric-graph loop soups on $\mathbb{Z}^d$ and $\widetilde{\mathbb{Z}}^d$ within a unified framework. For the continuum model, when the one-arm exponent $ξ(α)$ is positive, we prove that almost surely every cluster h