β-Skewed Maximal Spanning Forests
Abstract
The Free w-Maximal Spanning Forest (FMaxSF) is a weighted generalization of the classical Free Minimal Spanning Forest (FMSF) that is able to detect nonhyperfiniteness in percolation on nonunimodular graphs. We introduce a parameterized family of invariant random spanning forests that interpolates between these models. For every finite positive value of the parameter β, the construction retains many of the desired properties of FMaxSF while also admitting the finite-subtree forcing property of FMSF. We study local limits of these forests and, in particular, show that the small-β limit of the wired variant coincides with FMSF if and only if ph=pu, where ph is the threshold for the existence of heavy clusters and pu is the uniqueness threshold for Bernoulli(p) percolation. Finally, we show that the Free and the Wired w-Maximal Spanning Forests may coincide even if ph<pu, providing a negative answer to a question of Terlov and Timár.
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
- A zero-one law for swap-invariant events in uniform spanning forests
- Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations
BibTeX
@misc{chatterjee2026βskewed,
title = {{$β$-Skewed Maximal Spanning Forests}},
author = {Shirshendu Chatterjee and Yang Chen and Grigory Terlov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11845},
eprint = {2609.11845},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11845},
}
RIS
TY - UNPB
TI - β-Skewed Maximal Spanning Forests
AU - Shirshendu Chatterjee
AU - Yang Chen
AU - Grigory Terlov
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11845
UR - https://arxiv.org/abs/2609.11845
AB - The Free w-Maximal Spanning Forest (FMaxSF) is a weighted generalization of the classical Free Minimal Spanning Forest (FMSF) that is able to detect nonhyperfiniteness in percolation on nonunimodular graphs. We introduce a parameterized family of invariant random spanning forests that interpolates between these models. For every finite positive value of the parameter β, the construction retains many of the desired properties of FMaxSF while also admitting the finite-subtree forcing property of FMSF. We study local limits of these forests and, in particular, show that the small-β limit of the wired variant coincides with FMSF if and only if p_h=p_u, where p_h is the threshold for the existence of heavy clusters and p_u is the uniqueness threshold for Bernoulli(p) percolation. Finally, we show that the Free and the Wired w-Maximal Spanning Forests may coincide even if p_h<p_u, providing a negative answer to a question of Terlov and Timár.
ER -