Probability and statistical mechanics · Result 214

The Benjamini–Schramm nonuniqueness conjecture

1 paperLean

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.

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Papers 1

Nonuniqueness of percolation on nonamenable quasi-transitive graphs

OpenAI · September 24, 2026

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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@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}
}

What was formalized in Lean

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

Scope note on GitHub →