Result 214, Probability and statistical mechanics

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.

Lean formalization Proof

The bigger picture

Why it matters

The manuscript claims that randomly retaining edges in certain infinite networks produces a nonempty probability range with infinitely many infinite connected regions, almost surely. Coexistence, rather than immediate formation of one giant component, is therefore an unavoidable phase.

What changes?

The claim covers every infinite, connected graph with finitely many neighbors per vertex, finitely many vertex types under symmetry, and a uniform positive lower bound on boundary size relative to the size of finite vertex sets. These are the locally finite, quasi-transitive and nonamenable assumptions. In Bernoulli bond percolation, each edge is retained independently with probability p. The threshold for an infinite cluster is strictly below the threshold for a unique one.

What does that help mathematicians do?

The stronger bound places the threshold for boundedness of the connection-probability operator above the emergence threshold and at or below the uniqueness threshold. This operator acts on vertex weights with finite sum of squares, weighting pairs by their connection probability. The bound also establishes the critical triangle condition: a finiteness test for sums of products of three connection probabilities at the emergence threshold. This supplies quantitative control over connectivity, not just a count of infinite clusters.

Are there practical applications?

Its immediate value is foundational for probability and statistical mechanics: it separates the onset of unbounded connectivity from the onset of uniqueness on these graphs. The result concerns infinite-network models, not a tested procedure for finite networks; any use in concrete network modeling would require additional justification.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Nonuniqueness of percolation on nonamenable quasi-transitive graphs

September 24, 2026 53 pages

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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Cite (BibTeX)

@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's note on what the formalization covers, from lean/docs/214.md.

The Benjamini–Schramm nonuniqueness conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Nonuniqueness of percolation on nonamenable quasi-transitive graphs

Scope

The formalization proves the nonuniqueness phase conjecture for Bernoulli bond percolation on infinite connected locally finite nonamenable quasi-transitive graphs. It establishes $p_c<p_{2\to2}\le p_u$ and a common nonempty coupled interval with infinitely many infinite clusters almost surely. For Cayley graphs, the result holds for every finite symmetric generating set of a nonamenable finitely generated group and every $p\in(p_c,p_u)$ .

The selected critical estimates include a finite triangle diagram, susceptibility of order $(p_c-p)^{-1}$ , percolation probability of order $p-p_c$ , cluster-volume tail of order $n^{-1/2}$ , and intrinsic and extrinsic radius tails of order $n^{-1}$ . Connection probabilities decay exponentially below $p_{2\to2}$ .

Comparator links

Result	Comparator statement
Nonuniqueness phase and critical laws on quasi-transitive graphs	BenjaminiSchramm.lean
Nonuniqueness for every nonamenable Cayley graph	CayleyPercolation.lean

At a glance

Subject: Probability and statistical mechanics
Manuscripts: 1
Pages: 53
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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