On loop-soup clusters in three dimensions and higher
Abstract
We study clusters of the Brownian loop soup on ℝ3 and of random walk and metric-graph loop soups on ℤd and ℤ̃d 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↓0. 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 ℤd and ℤ̃d for d=3,4,5, yielding capacity lower bounds of order r, r2/log r, and r2 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 ℤ̃d has upper endpoint 1/2 by adapting the OSSS sharpness framework to the truncated loop soup.
Related
- Scaling Limit of Critical Loop Soup Clusters in Three, Four, and Five Dimensions
- Vanishing orders of Dirichlet solutions to the Schrödinger equation in dimensions three and higher
- Loop soup percolation and boundary dimension on metric trees
- The Hörmander-Bernhardsson function in higher dimensions
- Decay estimates for Yang-Mills connections and Yang-Mills-Higgs pairs in higher dimensions
BibTeX
@misc{gao2026loopsoup,
title = {{On loop-soup clusters in three dimensions and higher}},
author = {Yifan Gao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38510},
eprint = {2610.38510},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38510},
}
RIS
TY - UNPB
TI - On loop-soup clusters in three dimensions and higher
AU - Yifan Gao
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38510
UR - https://arxiv.org/abs/2610.38510
AB - We study clusters of the Brownian loop soup on ℝ^3 and of random walk and metric-graph loop soups on ℤ^d and ℤ̃^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↓0. 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 ℤ^d and ℤ̃^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 ℤ̃^d has upper endpoint 1/2 by adapting the OSSS sharpness framework to the truncated loop soup.
ER -