The Benjamini-Schramm criticality conjecture for bond percolation on quasi-transitive graphs

Let $G$ be an infinite, connected, locally finite quasi-transitive graph (its automorphism group has finitely many vertex orbits) and run Bernoulli percolation on $G$ with critical probability $p_c<1$ . Benjamini and Schramm (1996, Conjecture 4) conjectured that critical percolation dies on every such graph: at $p=p_c$ there is almost surely no infinite open cluster. Their paper discusses site percolation and says the questions remain equally valid for bond percolation outside planar settings. The motivating case is $\mathbb Z^d$ , where $\theta(p_c)=0$ was known for $d=2$ and in high dimensions only. Does critical Bernoulli bond percolation have no infinite cluster on every infinite connected locally finite quasi-transitive graph with $p_c<1$ ?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Argument
Field: Percolation theory on graphs and groups
Posed by: Itai Benjamini and Oded Schramm, Percolation beyond Z^d, many questions and a few answers (1996), Conjecture 4
Year posed: 1996
Years open: 30y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 79 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims that for every infinite connected locally finite quasi-transitive graph with $p_c<1$ , Bernoulli bond percolation at $p_c$ has almost surely no infinite cluster. Exponential growth is Hutchcroft's theorem; the new work covers all subexponential growth, split into superpolynomial growth (a two-cluster entropy argument) and growth polynomial along a sequence of radii (a nilpotent-quotient corridor exploration). It includes $\theta(p_c)=0$ on $\mathbb Z^d$ for every $d\ge2$ . The companion proves critical bond and site nonpercolation on $\mathbb Z^3$ . It does NOT settle the site form of the conjecture on general quasi-transitive graphs, and gives no critical exponents or quantitative decay.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. Both manuscripts are credited to OpenAI with no human author named. The quasi-transitive manuscript credits earlier public AI-produced work on the Kozma-Nitzan gluing inequality (Leder's Lean account for Z^d, and expositions prompted by Ahmed Bou-Rabee) and gives its own proof of the gluing input.

Verification

No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript was read against Benjamini-Schramm Conjecture 4 (the source was opened): it is the conjecture for bond percolation, with no extra hypothesis beyond $p_c<1$ . The Lean challenge ComparatorChallenges/CriticalPercolation.json (theorem OAI.CriticalPercolation.BondGraph.no_percolation_at_criticality, solution module OAI.Probability.CriticalPercolation.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/213.md. Its statement was read here: for a bond graph with possibly parallel bonds and loops, infinite vertex set, connected, locally finite counting bonds, finitely many vertex orbits under automorphisms, and critical probability below one, the Bernoulli product law at $p_c$ gives probability zero to the event that some cluster is infinite. That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The companion's Z^3 bond and site statement (CriticalZ3, listed in formalization.yaml) was also read and matches its abstract. The proof uses Hutchcroft's 2016 theorem for exponential growth and the Tessera-Tointon structure theorem as cited inputs.

The Benjamini-Schramm criticality conjecture for bond percolation on quasi-transitive graphs

What was actually shown

What the AI did

Verification

Sources

Discussion