Short··0:12·Probability and statistical mechanics
Critical Percolation on a Cubic Lattice
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About this video
When do random connections form a giant network? OpenAI's manuscript, Critical bond and site percolation on the cubic lattice, reports no infinite cluster at the respective critical probabilities on Z^3. The paper acknowledges earlier AI-produced work prompted by Ahmed Bou-Rabee and public expositions by Leder and Kozma-Nitzan. This animation varies connection probability on a finite cubic grid; it does not display an infinite lattice or identify the exact critical probability.
About the animation: A seeded sample of independent uniform edge thresholds on a 10 by 10 by 10 cubic grid gives open bonds when the threshold is below p. The animation sweeps p from 0.12 to 0.42 and back, using monotone coupling. Union-find determines the actual connected components. Gold identifies the largest component; it can span this finite box. A finite spanning cluster does not contradict the reported absence of an infinite cluster at the infinite-lattice critical probability. The video shows bond percolation only; the paper also treats site percolation.
Paper: github.com/openai/math/blob/main/preprints/Critical-bond-and-…
GitHub materials: github.com/openai/math/blob/main/lean/docs/213.md
Repository: github.com/openai/math
Source-reported result. This visual explanation is not an independent proof audit or a verification of the formalization.
'Titan' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
scottbuckley.com.au/library/titan/
Licence: creativecommons.org/licenses/by/4.0/
Music edits: Excerpted, crossfaded to loop, volume adjusted.
About the problem
Probability and statistical mechanics·OpenAI Math·Result 213
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 < 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)
Topics
Critical bond and site percolation on the cubic latticecritical percolationbond percolationsite percolationcubic latticeinfinite clusterprobability theoryphase transitionrandom graphsOpenAI math3D math animation
