Probability and statistical mechanics · Result 213

Critical percolation on every quasi-transitive graph

2 papersLean

Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with \(p_c\lt 1\): at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ³.

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The problem

What is being asked, where it comes from, and why it is hard. No proof needed.

▶Introduction to the problemin preparation

The solution proposed by OpenAI

How the paper solves it, step by step.

Full explanation

▶The proposed solutionin preparation

Papers 2

Critical bond and site percolation on the cubic lattice

OpenAI · September 24, 2026

We prove that nearest-neighbor Bernoulli bond and site percolation on ℤ³ have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.

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Cite

@misc{OAI:Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026,
  author = {{OpenAI}},
  title = {{Critical bond and site percolation on the cubic lattice}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf}{OAI:Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026}},
  year = {2026}
}

No percolation at criticality on quasi-transitive graphs

OpenAI · September 24, 2026

We prove that critical Bernoulli bond percolation has no infinite cluster on any infinite connected locally finite quasi-transitive graph with critical probability less than one. This resolves the bond form of the criticality conjecture of Benjamini and Schramm.

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Cite

@misc{OAI:No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026,
  author = {{OpenAI}},
  title = {{No percolation at criticality on quasi-transitive graphs}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026/paper.pdf}{OAI:No-percolation-at-criticality-on-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):

Scope

The critical-percolation question asks whether an infinite cluster can remain at the threshold. The formalized result proves that, at their respective critical probabilities, nearest-neighbor bond and site percolation on \(\mathbb Z^3\) almost surely have no infinite cluster. Equivalently, almost surely every vertex belongs to a finite cluster in each model.

The formalization proves absence of an infinite cluster at criticality for Bernoulli bond percolation on every infinite connected locally finite quasi-transitive graph with critical probability \(p_c<1\). At \(p=p_c\), the probability that any infinite cluster exists is zero. The graph model permits bond multiplicities.

Comparator links

Result	Comparator statement
No infinite critical clusters on \(\mathbb Z^3\)	CriticalZ3.lean
No infinite cluster at the critical probability	CriticalPercolation.lean

Scope note on GitHub →