Result 213, Probability and statistical mechanics

Critical percolation on every quasi-transitive graph

Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with pc<1p_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.

Lean formalization Proof

The bigger picture

Why it matters

Percolation studies when random local connections create an infinite connected network. These manuscripts claim that, across a broad class of symmetric networks, the exact transition point still has only finite connected pieces.

What changes?

In Bernoulli bond percolation, each edge is independently kept with probability p; clusters are vertices connected by kept edges. The critical probability is the threshold for an infinite cluster. The manuscript reports that, at this threshold, almost surely no infinite cluster exists on every infinite connected graph that is locally finite (each vertex has finitely many neighbors) and quasi-transitive (its symmetries give finitely many vertex types), provided the critical probability is less than one.

What does that help mathematicians do?

Consequently, at criticality the probability that a fixed vertex connects beyond distance r tends to zero as r grows. This gives researchers a qualitative constraint on long-distance connectivity, without specifying a decay rate. The companion manuscript reports the same absence of infinite clusters for nearest-neighbor bond and site percolation on the three-dimensional cubic lattice, at their respective thresholds; site percolation independently retains vertices instead of edges.

Are there practical applications?

The immediate value is foundational for probability and statistical mechanics: these results characterize whether an infinite connected phase exists at the transition itself. Percolation provides a mathematical framework for random connectivity. The reported theorems settle this qualitative boundary question in the stated settings, rather than providing numerical thresholds or practical simulation methods.

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.

2 manuscripts

Critical bond and site percolation on the cubic lattice

September 24, 2026 20 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 42 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/213.md.

Critical percolation on every quasi-transitive graph

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 Z3\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 pc<1p_c<1. At p=pcp=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 Z3\mathbb Z^3 CriticalZ3.lean
No infinite cluster at the critical probability CriticalPercolation.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.