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Nonuniqueness of percolation on nonamenable quasi-transitive graphs
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Abstract
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 $p_c\lt p_{2\to2}\le p_u$.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Tasmin Chu, reaction in 100+ reactions to 100+ solutions, Proofs and Prompts (blog post of 2026-10-08, reactions updated to 2026-10-19). Recalls the 1996 Benjamini–Schramm conjecture and remarks: "Unsurprisingly the OpenAI preprint builds on work of my advisor Tom Hutchcroft", also describing the preprint's impact on Chu's own research plans.
- Inhyeok Choi, reaction in 100+ reactions to 100+ solutions, Proofs and Prompts (2026-10-08). Choi, co-author of the cited Choi–Seo paper on acylindrically hyperbolic groups, says they had thought about the $p_c<p_u$ question and asks "whether there is a more natural proof for groups with free subgroups."
Formalization
- OpenAI math release, The Benjamini–Schramm nonuniqueness conjecture: Lean scope note (lean/docs/214.md), mirrored on github.com/mathcommunity/oai-math-revised (release of 2026-10-06). Describes the release's own Lean formalization of $p_c<p_{2\to2}\le p_u$, the coupled nonuniqueness interval, the arbitrary-generating-set Cayley-graph corollary and selected critical and exponential-connectivity estimates, with comparator files
BenjaminiSchramm.leanandCayleyPercolation.lean. [family-level]
Related prior literature mentioned by others
- Tom Hutchcroft, The $L^2$ boundedness condition in nonamenable percolation (arXiv:1904.05804, 2019; revised 2020-09-23). Linked by the BinaryVerse AI article to explain how the operator condition yields more than nonuniqueness, including exponential connectivity and critical-radius estimates; also in the paper's bibliography.
Possibly relevant literature (found by the agent, not checked)
- Russell Lyons, Fixed price of groups and percolation (arXiv:1109.5418; Ergodic Theory Dynam. Systems 33 (2013) 183–185). Shows that a finitely generated group either has fixed price or every one of its Cayley graphs has a Bernoulli percolation phase with infinitely many infinite clusters, an earlier all-generating-sets result relevant to Corollary 1.2 (arbitrary Cayley graphs).
- Zhongyang Li, Tree embeddings and nonuniqueness in site percolation (arXiv:2304.00923, v7 2026-07-05). Proves nonuniqueness for Bernoulli site percolation on properly embedded planar graphs of minimum degree at least 7; relevant since Benjamini–Schramm's standing model is site percolation, while Theorem 1.1 concerns bond percolation.
- Tom Hutchcroft and Minghao Pan, Percolation at the uniqueness threshold via subgroup relativization (arXiv:2409.12283, 2024-09-18). Its nonuniqueness results at $p_u$ for specific nonamenable groups give endpoint context for Theorem 1.1's strict gap $p_c<p_u$, which does not itself determine the cluster count at $p_u$.
Digests and expositions
- Latent Seminar, Nonamenable percolation and the two-point operator (2026-10-06). Exposition of the connection operator, corridor deformation, spectral contradiction and simultaneous coupling argument; it states that it does not independently verify the complete proofs or audit the Lean development. [family-level]
News and blogs
- Gil Kalai, Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans (Combinatorics and more, 2026-10-07). Item 20 ("Percolation: criticality and nonuniqueness") lists the release's claim of $p_c<p_u$ for nonamenable quasi-transitive graphs and links Kalai's 2009 discussion of the Benjamini–Schramm conjecture.
- Azmat, Benjamini–Schramm Nonuniqueness Conjecture: OpenAI Proof Cracks the 30-Year Mystery of Infinite Clusters (BinaryVerse AI, 2026-10-14). Describes the paper's graph hypotheses, operator gap, proof strategy and critical-law consequences, distinguishing the asserted interval from behavior exactly at $p_u$.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 214
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026,
author = {{OpenAI}},
title = {{Nonuniqueness of percolation on nonamenable quasi-transitive graphs}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf}{OAI:Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/214.md. The Lean statements have not been checked by people against the paper's claims.
Family 214: The Benjamini–Schramm nonuniqueness conjecture
Proves $p_c\lt p_u$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.
This is the only paper in this family.