Non-perturbative finite connectivities in supercritical Bernoulli percolation on ℤd with d≥ 3
Abstract
We solve a longstanding problem in Bernoulli percolation: we derive sharp Ornstein--Zernike asymptotics for the probability that two distant points are connected by a finite cluster in the whole supercritical regime. Our proof has two main ingredients. First, the renewal structure from Fridbergh and Hammond (2024) yields an a priori bound on cluster volume. Second, we adapt the coarse-graining and irreducible-cluster framework of Companion, Ioffe, Velenik (2008) to finite supercritical connections. The main distinction from the subcritical analysis is that our argument does not require a separation of masses between irreducible and unrestricted finite connections. The volume bound and a modified coarse-graining construction instead provide stretched-exponential tails for the effective random-walk increments, which suffice to obtain the sharp asymptotics.
Related
- Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters
- A finite-graph conjecture related to θ(pc)=0 for Bernoulli bond percolation on ℤd
- Supercritical level-set percolation on finite graphs
- Locality of Bernoulli Site Percolation on Transitive Graphs
- Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
BibTeX
@misc{aoun2026nonperturbative,
title = {{Non-perturbative finite connectivities in supercritical Bernoulli percolation on $\mathbbZ^d$ with $d\geq 3$}},
author = {Yacine Aoun and Kamil Khettabi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08682},
eprint = {2610.08682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08682},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-perturbative finite connectivities in supercritical Bernoulli percolation on ℤ^d with d≥ 3
AU - Yacine Aoun
AU - Kamil Khettabi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08682
UR - https://arxiv.org/abs/2610.08682
AB - We solve a longstanding problem in Bernoulli percolation: we derive sharp Ornstein--Zernike asymptotics for the probability that two distant points are connected by a finite cluster in the whole supercritical regime. Our proof has two main ingredients. First, the renewal structure from Fridbergh and Hammond (2024) yields an a priori bound on cluster volume. Second, we adapt the coarse-graining and irreducible-cluster framework of Companion, Ioffe, Velenik (2008) to finite supercritical connections. The main distinction from the subcritical analysis is that our argument does not require a separation of masses between irreducible and unrestricted finite connections. The volume bound and a modified coarse-graining construction instead provide stretched-exponential tails for the effective random-walk increments, which suffice to obtain the sharp asymptotics.
ER -