OP-00700 Solution claimed

A nonuniqueness phase on every nonamenable transitive graph

Let $G = (V,E)$ be an infinite, connected, locally finite, vertex-transitive graph which is nonamenable: $$\inf_{\varnothing \ne K \subset V,\ |K| < \infty} \frac{|\partial_E K|}{|K|} > 0,$$ where $\partial_E K$ is the set of edges with exactly one endpoint in $K$. For Bernoulli bond percolation on $G$ let $$p_c = \inf\{p : \mathbb{P}_p(\text{an infinite open cluster exists}) > 0\}, \qquad p_u = \inf\{p : \mathbb{P}_p(\text{exactly one infinite open cluster exists}) = 1\}.$$ On an amenable transitive graph there is almost surely at most one infinite cluster (Burton and Keane; Gandolfi, Keane and Newman), so that $p_c = p_u$. Pak and Smirnova-Nagnibeda proved that every nonamenable finitely generated group has some Cayley graph with $p_c < p_u$, and Hutchcroft proved $p_c < p_u$ for every nonamenable, Gromov hyperbolic, quasi-transitive graph.

Problem. Must $p_c < p_u$?

Source: Benjamini and Schramm (1996), Conjecture 6, for quasi-transitive graphs; Lyons and Peres (2016), Conjecture 7.31. Stated by AIM — Open Applied Problems for vertex-transitive graphs.

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Secondary: GR, MP
Keywords: AIM Open Applied Problems, nonamenable graphs, percolation, uniqueness threshold

References

@article{benjamini1996,
  author  = {Benjamini, Itai and Schramm, Oded},
  title   = {Percolation beyond {$\mathbb{Z}^d$}, many questions and a few answers},
  journal = {Electron. Comm. Probab.},
  volume  = {1},
  number  = {8},
  year    = {1996},
  pages   = {71--82},
  doi     = {10.1214/ECP.v1-978},
}
@book{lyons2016,
  author    = {Lyons, Russell and Peres, Yuval},
  title     = {Probability on Trees and Networks},
  series    = {Cambridge Series in Statistical and Probabilistic Mathematics},
  volume    = {42},
  publisher = {Cambridge University Press},
  address   = {New York},
  year      = {2016},
  doi       = {10.1017/9781316672815},
}
@article{hutchcroft2019,
  author        = {Hutchcroft, Tom},
  title         = {Percolation on hyperbolic graphs},
  journal       = {Geom. Funct. Anal.},
  volume        = {29},
  number        = {3},
  year          = {2019},
  pages         = {766--810},
  doi           = {10.1007/s00039-019-00498-0},
  eprint        = {1804.10191},
  archivePrefix = {arXiv},
}
@misc{aim-nonamenable-nonuniqueness-phase,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A nonuniqueness phase on every nonamenable transitive graph}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/176-nonamenable-nonuniqueness-phase.md},
  note         = {Entry 176 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Nonuniqueness of percolation on nonamenable quasi-transitive graphs v1 GitHub ↗

OpenAI

We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing $p_c<p_{2\to2}\le p_u$.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00700, "A nonuniqueness phase on every nonamenable transitive graph", clopen.solutions.