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

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Benjamini-Schramm criticality conjecture for bond percolation on quasi-transitive graphs · Benjamini-Schramm criticality (bond form)

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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Named evidence

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

specialist registry · passed

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.

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Companion: Critical bond and site percolation on the cubic lattice

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paper link supplied by the VibeMathed registry.

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Lean: solution module for the quasi-transitive theorem

lean artifact · pending