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 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. (Lean)

Manuscripts (2)

Critical bond and site percolation on the cubic lattice

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:Critical-bond-and-site-percolation-on-the-cubic-lattice-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.

No percolation at criticality on quasi-transitive graphs

September 24, 2026 · source · cite as \cite{OAI:No-percolation-at-criticality-on-quasi-transitive-graphs-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.

Result 213 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.