No infinite critical clusters on quasi-transitive graphs at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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No infinite critical clusters on quasi-transitive graphs

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	luck, magnets	Levels:	2
Category:	Probability and statistical mechanics	Lean version:	YES! ✔
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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.

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LEVEL 1 Critical bond and site percolation on the cubic lattice

released 2026-09-24 | 1 theorem · 9 lemmas · 11 proofs · 9,429 words | PLAY LEVEL 1 » (pdf)

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.

LEVEL 2 No percolation at criticality on quasi-transitive graphs

released 2026-09-24 | 5 theorems · 10 lemmas · 20 proofs · 20,076 words | PLAY LEVEL 2 » (pdf)

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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