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 has Hausdorff dimension $3-ξ(α)$. A key step is to show that, conditioned on the one-arm event, a crossing cluster has capacity comparable to the crossing scale with uniformly positive probability. This gives one-arm estimates of order $αr^{-ξ(α)}$, uniformly down to zero intensity, and shows that $ξ(α)$ is strictly decreasing and left-continuous. Moreover, $1-ξ(α)\asympα$ as $α\downarrow0$. As another application of the one-arm estimates, we prove that several critical values coincide, resolving a conjecture of Jego and Lupu in arXiv:2601.04840. The proof combines a decoupling technique with a multiscale recursive argument to control the combined effects of large and small loops. These methods also extend to loop soups on $\mathbb{Z}^d$ and $\widetilde{\mathbb{Z}}^d$ for $d=3,4,5$, yielding capacity lower bounds of order $r$, $r^2/\log r$, and $r^2$ for the cluster of the origin, conditioned on the one-arm event, throughout the small-loop decay regime. In particular, these bounds imply strict polynomial comparisons between one-arm probabilities at different intensities in dimension three. We also show that the small-loop decay regime on $\widetilde{\mathbb{Z}}^d$ has upper endpoint $\frac{1}{2}$ by adapting the OSSS sharpness framework to the truncated loop soup.