On the one-arm probability for the metric graph Gaussian free field in high dimensions
Abstract
We consider the sign clusters of the Gaussian free field on the cable system of a transient weighted graph with volume growth exponent α and Green function decay exponent ν. We prove that the probability that the cluster of a vertex reaches distance R is of order R−β when 2α<3ν, where β=α−ν is the walk dimension, and that the same holds up to a logarithmic factor in the upper bound when 2α=3ν. On ℤd these conditions read d>6 and d=6. The upper bound combines a general entropic bound, which compares the law of the cluster under the critical loop soup and under a massive one, with a Simon-Lieb-type inequality for the massive field derived from its Markov property. The arguments are robust and require no transitivity. The lower bound follows from a competition between the capacity and the volume of the cluster. In both bounds, the key quantity is the expected exit time of the random walk from balls, which links the volume of the cluster to its radius.
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BibTeX
@misc{rodriguez2026onearm,
title = {{On the one-arm probability for the metric graph Gaussian free field in high dimensions}},
author = {Pierre-François Rodriguez and Jialu Shi and Wen Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05357},
eprint = {2611.05357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05357},
}
RIS
TY - UNPB
TI - On the one-arm probability for the metric graph Gaussian free field in high dimensions
AU - Pierre-François Rodriguez
AU - Jialu Shi
AU - Wen Zhang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05357
UR - https://arxiv.org/abs/2611.05357
AB - We consider the sign clusters of the Gaussian free field on the cable system of a transient weighted graph with volume growth exponent α and Green function decay exponent ν. We prove that the probability that the cluster of a vertex reaches distance R is of order R^−β when 2α<3ν, where β=α−ν is the walk dimension, and that the same holds up to a logarithmic factor in the upper bound when 2α=3ν. On ℤ^d these conditions read d>6 and d=6. The upper bound combines a general entropic bound, which compares the law of the cluster under the critical loop soup and under a massive one, with a Simon-Lieb-type inequality for the massive field derived from its Markov property. The arguments are robust and require no transitivity. The lower bound follows from a competition between the capacity and the volume of the cluster. In both bounds, the key quantity is the expected exit time of the random walk from balls, which links the volume of the cluster to its radius.
ER -