PreprintarXiv licencearXiv
A zero-one law for swap-invariant events in uniform spanning forests
Abstract
We prove that every event invariant under swaps has probability zero or one for either the free or the wired spanning forest on any connected, locally finite, infinite network. Applications include the constancy of the number of components and of the total number of excessive ends. For the free forest, these results resolve longstanding questions of Benjamini, Lyons, Peres, and Schramm [Ann. Probab. (2001), Question 15.7] on constancy of the component count and of Lyons, Morris, and Schramm [Electron. J. Probab. (2008), Question 7.7] on the zero-one law for one-endedness. In the latter case, we obtain the stronger conclusion that the total number of excessive ends is deterministic.
Related
- Spanning Even Linear Forests with Isolated Edges Share the Anti-Ramsey Numbers of Perfect Matchings
- Sharp Bounds for Common Spanning Forests under Color Refinement
- A sharp upper bound on the number of spanning forests of regular graphs
- β-Skewed Maximal Spanning Forests
- Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations
BibTeX
@misc{li2026zeroone,
title = {{A zero-one law for swap-invariant events in uniform spanning forests}},
author = {Xinyi Li and Yao Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00938},
eprint = {2610.00938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00938},
}
RIS
TY - UNPB
TI - A zero-one law for swap-invariant events in uniform spanning forests
AU - Xinyi Li
AU - Yao Yu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00938
UR - https://arxiv.org/abs/2610.00938
AB - We prove that every event invariant under swaps has probability zero or one for either the free or the wired spanning forest on any connected, locally finite, infinite network. Applications include the constancy of the number of components and of the total number of excessive ends. For the free forest, these results resolve longstanding questions of Benjamini, Lyons, Peres, and Schramm [Ann. Probab. (2001), Question 15.7] on constancy of the component count and of Lyons, Morris, and Schramm [Electron. J. Probab. (2008), Question 7.7] on the zero-one law for one-endedness. In the latter case, we obtain the stronger conclusion that the total number of excessive ends is deterministic.
ER -