Critical one-arm probabilities for metric-graph loop soups in dimensions d≥3
Abstract
We study the critical Brownian loop soup on the metric graph ℤ̃d. Using a new unified approach, we determine the one-arm probability up to multiplicative constants in all dimensions d≥3. In particular, at the upper critical dimension dc=6, we show that the one-arm probability is comparable to r−2(log r)1/2, and moreover, with high probability a box of radius r contains on the order of log r macroscopic clusters, each with volume of order r4(log r)−1/2. For d≠6, we recover the one-arm estimates of Cai and Ding, r−(d−2)/2 for d=3,4,5 and r−2 for d>6 (arXiv:2406.02397; Probab. Theory Relat. Fields, 2025), and the three-dimensional estimate of Drewitz, Prévost and Rodriguez (Probab. Theory Relat. Fields, 2026). Our proof adapts the entropy method of Dewan and Muirhead (Probab. Theory Relat. Fields, 2023) to the loop-soup setting. This yields a renormalization inequality for the one-arm probability, from which we derive the upper bounds. Independently, Rodriguez, Shi, and Zhang (arXiv:2611.05357) have recently used a closely related entropy approach to recover the one-arm estimates for d>6 and improve the upper bound for d=6. They also announce the same sharp one-arm estimate for d=6 in forthcoming work.
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BibTeX
@misc{gao2026critical,
title = {{Critical one-arm probabilities for metric-graph loop soups in dimensions $d\ge3$}},
author = {Yifan Gao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06298},
eprint = {2611.06298},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06298},
}
RIS
TY - UNPB
TI - Critical one-arm probabilities for metric-graph loop soups in dimensions d≥3
AU - Yifan Gao
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06298
UR - https://arxiv.org/abs/2611.06298
AB - We study the critical Brownian loop soup on the metric graph ℤ̃^d. Using a new unified approach, we determine the one-arm probability up to multiplicative constants in all dimensions d≥3. In particular, at the upper critical dimension d_c=6, we show that the one-arm probability is comparable to r^−2(log r)^1/2, and moreover, with high probability a box of radius r contains on the order of log r macroscopic clusters, each with volume of order r^4(log r)^−1/2. For d≠6, we recover the one-arm estimates of Cai and Ding, r^−(d−2)/2 for d=3,4,5 and r^−2 for d>6 (arXiv:2406.02397; Probab. Theory Relat. Fields, 2025), and the three-dimensional estimate of Drewitz, Prévost and Rodriguez (Probab. Theory Relat. Fields, 2026). Our proof adapts the entropy method of Dewan and Muirhead (Probab. Theory Relat. Fields, 2023) to the loop-soup setting. This yields a renormalization inequality for the one-arm probability, from which we derive the upper bounds. Independently, Rodriguez, Shi, and Zhang (arXiv:2611.05357) have recently used a closely related entropy approach to recover the one-arm estimates for d>6 and improve the upper bound for d=6. They also announce the same sharp one-arm estimate for d=6 in forthcoming work.
ER -