Loop soup percolation and boundary dimension on metric trees
Abstract
We determine the percolation threshold of the Brownian loop soup on a transient locally finite weighted tree with proper cable metric and no killing. It equals 1−DG/2, where DG∈[0,2] is the Hausdorff dimension of the boundary in the metric induced by the normalized Green function. This gives an analogue of Lyons' branching-number formula for Bernoulli percolation, with Green distance replacing graph distance. Spherically symmetric trees with unit conductances and branching number two realize every threshold in [1/2,1], while geometric conductances give explicit positive thresholds below one-half and a subdivided binary tree has threshold zero. Under the capacity assumption of Drewitz, Prévost and Rodriguez, we characterize the threshold one-half by a finite-energy condition, without a minimum-degree assumption. Non-percolation at one-half then follows from their theorem and Lupu's coupling. For supercritical Galton-Watson trees with unit conductances, conditioned on survival, the boundary dimension is almost surely one and the threshold is one-half, with non-percolation at criticality. No offspring moment assumption is required, and leaves and vertices with one child are allowed. The classification also applies intrinsically to proper real trees spanned by their infinite rays.
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BibTeX
@misc{monch2026loop,
title = {{Loop soup percolation and boundary dimension on metric trees}},
author = {Christian Mönch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12053},
eprint = {2610.12053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Loop soup percolation and boundary dimension on metric trees
AU - Christian Mönch
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12053
UR - https://arxiv.org/abs/2610.12053
AB - We determine the percolation threshold of the Brownian loop soup on a transient locally finite weighted tree with proper cable metric and no killing. It equals 1−D_G/2, where D_G∈[0,2] is the Hausdorff dimension of the boundary in the metric induced by the normalized Green function. This gives an analogue of Lyons' branching-number formula for Bernoulli percolation, with Green distance replacing graph distance. Spherically symmetric trees with unit conductances and branching number two realize every threshold in [1/2,1], while geometric conductances give explicit positive thresholds below one-half and a subdivided binary tree has threshold zero. Under the capacity assumption of Drewitz, Prévost and Rodriguez, we characterize the threshold one-half by a finite-energy condition, without a minimum-degree assumption. Non-percolation at one-half then follows from their theorem and Lupu's coupling. For supercritical Galton-Watson trees with unit conductances, conditioned on survival, the boundary dimension is almost surely one and the threshold is one-half, with non-percolation at criticality. No offspring moment assumption is required, and leaves and vertices with one child are allowed. The classification also applies intrinsically to proper real trees spanned by their infinite rays.
ER -