Does supercritical percolation preserve transience?
Let $G$ be the Cayley graph of an infinite finitely generated group with respect to a finite symmetric generating set, and suppose that simple random walk on $G$ is transient. Let $p_c(G)$ be the critical probability of Bernoulli bond percolation on $G$: the infimum of the $p$ for which, keeping each edge independently with probability $p$, there is an infinite cluster with positive probability. Grimmett, Kesten and Zhang proved that for $d \ge 3$ simple random walk on the infinite cluster of supercritical percolation on $\mathbb{Z}^d$ is transient. Benjamini, Lyons and Schramm proved that on a nonamenable Cayley graph simple random walk on an infinite cluster has positive speed, so the infinite clusters are transient. Hutchcroft proved that on every transient transitive graph the infinite clusters are transient for all $p < 1$ close enough to $1$, and that on a transitive graph of polynomial growth of dimension $d$ they have anchored isoperimetric dimension $d$ for every $p > p_c$; graphs of anchored isoperimetric dimension greater than $2$ are transient (Thomassen).
Problem. For every such $G$ and every $p > p_c(G)$, is simple random walk on every infinite cluster of Bernoulli($p$) bond percolation on $G$ transient, almost surely?
Source: Benjamini, Lyons and Schramm (1999), Conjecture 1.7, for Bernoulli percolation; see Lyons and Peres (2016), Conjecture 6.44. Stated by AIM — Open Applied Problems for bond percolation with p > p_c.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- GR
- Keywords
- AIM Open Applied Problems, Cayley graphs, percolation, random walks
References
@incollection{benjamini1999,
author = {Benjamini, Itai and Lyons, Russell and Schramm, Oded},
title = {Percolation perturbations in potential theory and random walks},
booktitle = {Random Walks and Discrete Potential Theory (Cortona, 1997)},
editor = {Picardello, M. and Woess, W.},
series = {Sympos. Math.},
volume = {XXXIX},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {1999},
pages = {56--84},
eprint = {math/9804010},
archivePrefix = {arXiv},
}
@book{lyons2016,
author = {Lyons, Russell and Peres, Yuval},
title = {Probability on Trees and Networks},
series = {Cambridge Series in Statistical and Probabilistic Mathematics},
publisher = {Cambridge University Press},
address = {New York},
year = {2016},
doi = {10.1017/9781316672815},
}
@article{grimmett1993,
author = {Grimmett, G. R. and Kesten, H. and Zhang, Y.},
title = {Random walk on the infinite cluster of the percolation model},
journal = {Probab. Theory Related Fields},
volume = {96},
number = {1},
year = {1993},
pages = {33--44},
doi = {10.1007/BF01195881},
}
@article{hutchcroft2023,
author = {Hutchcroft, Tom},
title = {Transience and anchored isoperimetric dimension of supercritical percolation clusters},
journal = {Electron. J. Probab.},
volume = {28},
year = {2023},
pages = {1--15},
doi = {10.1214/23-EJP905},
eprint = {2207.05226},
archivePrefix = {arXiv},
}
@article{thomassen1992,
author = {Thomassen, Carsten},
title = {Isoperimetric inequalities and transient random walks on graphs},
journal = {Ann. Probab.},
volume = {20},
number = {3},
year = {1992},
pages = {1592--1600},
doi = {10.1214/aop/1176989708},
}
@misc{aim-percolation-preserves-transience,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does supercritical percolation preserve transience?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/175-percolation-preserves-transience.md},
note = {Entry 175 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that for supercritical percolation on every infinite transitive graph, the probability that the origin belongs to a finite cluster of size at least $n$ decays exponentially in $Φ(n)$, where $Φ$ is the isoperimetric function of the graph.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2603.03257v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00403, "Does supercritical percolation preserve transience?", clopen.solutions.