A finite-graph conjecture related to θ(pc)=0 for Bernoulli bond percolation on ℤd
Abstract
We introduce a conjecture for Bernoulli bond percolation on finite graphs. Roughly speaking, it asserts that if each boundary vertex is associated with a highly probable event that is increasing with respect to the percolation configuration, then, conditionally on the origin being connected to the boundary, the origin is likely to be connected to a boundary vertex whose associated event occurs. The conjecture implies a high-probability connectivity statement which is known to imply θ(pc)=0 for Bernoulli bond percolation on ℤd, for every d≥2. We prove the conjecture for finite planar graphs when the origin and the boundary vertices lie on the outer-face boundary, with the explicit bound 1−2√ε. The proof combines a left-first depth-first exploration with a partial FKG inequality adapted to monotonicity up to a stopping time. A counterexample shows that the connectivity structure is essential: the analogous statement fails when the connectivity events are replaced by arbitrary increasing events.
Related
- Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
- Non-perturbative finite connectivities in supercritical Bernoulli percolation on ℤd with d≥ 3
- Locality of Bernoulli Site Percolation on Transitive Graphs
- Continuity of the phase transition for long-range Bernoulli percolation with exponential decay on ℤ2
- Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters
BibTeX
@misc{flammant2026finitegraph,
title = {{A finite-graph conjecture related to $θ(p_c)=0$ for Bernoulli bond percolation on $\mathbb Z^d$}},
author = {Lucas Flammant},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03492},
eprint = {2609.03492},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03492},
}
RIS
TY - UNPB
TI - A finite-graph conjecture related to θ(p_c)=0 for Bernoulli bond percolation on ℤ^d
AU - Lucas Flammant
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.03492
UR - https://arxiv.org/abs/2609.03492
AB - We introduce a conjecture for Bernoulli bond percolation on finite graphs. Roughly speaking, it asserts that if each boundary vertex is associated with a highly probable event that is increasing with respect to the percolation configuration, then, conditionally on the origin being connected to the boundary, the origin is likely to be connected to a boundary vertex whose associated event occurs. The conjecture implies a high-probability connectivity statement which is known to imply θ(p_c)=0 for Bernoulli bond percolation on ℤ^d, for every d≥2. We prove the conjecture for finite planar graphs when the origin and the boundary vertices lie on the outer-face boundary, with the explicit bound 1−2√ε. The proof combines a left-first depth-first exploration with a partial FKG inequality adapted to monotonicity up to a stopping time. A counterexample shows that the connectivity structure is essential: the analogous statement fails when the connectivity events are replaced by arbitrary increasing events.
ER -