Math
The "Percolation Phase Transition" Theorem Proven Beyond Lattices for All Symmetric Networks, Settling a 1996 Conjecture
Related terms: 確率 グラフ 相転移 指数関数 対称性
Key points (reviewing AI explanation)
In percolation theory, researchers proved that the transition in the probability of a "giant connected component" appearing when edges are randomly retained is sharp on the supercritical side as well, for all infinite transitive graphs. This was shown for lattices in the 1980s, with Benjamini and Schramm conjecturing the generalization in 1996. The proof, by Diskin, Easo, Radhakrishnan, Sudakov, and Tassion of ETH Zurich, was published as a preprint in March 2026 and reported by Quanta Magazine on August 31.
Explanation
Consider a sheet of grid paper where each line between squares is kept with probability p and removed with probability 1−p. While p is small, the regions connected by remaining lines are all small. Once p exceeds a certain value (the critical probability), a giant connected region suddenly appears that spans the entire sheet. This is the percolation phase transition, and it serves as the simplest model for phenomena such as hot water soaking into coffee grounds, forest fires spreading, and infectious diseases spreading.
In the 1980s, it was proven that, for lattices like grid paper, this transition is "sharp." If the probability is even slightly below the critical probability, the probability of connecting over long distances decays exponentially with distance. If it is even slightly above, exactly one giant connected component exists, and the probability of not connecting to it decays exponentially. The region before the critical point is called subcritical, and the region after is called supercritical.
In 1996, Itai Benjamini and Oded Schramm conjectured that "even for non-lattice networks, as long as they look the same from every point (transitive graphs) and extend infinitely, the same should hold true." The subcritical side was proven by Antunović and Veselić in 2007, but the supercritical side had only been proven using methods that relied on the properties of lattices, leaving the problem open for nearly 20 years.
Five researchers at ETH Zurich — Sahar Diskin, Philip Easo, Ananth Radhakrishnan, Benny Sudakov, and Vincent Tassion — proved the sharpness of the supercritical side for all infinite transitive graphs in a preprint published on March 3, 2026. They built their argument using only the graph's symmetry, without relying on lattice coordinates.
"What can be done for lattices can be done for all symmetric networks." The fundamental theorem of percolation has finally moved beyond the lattice.
Source: www.quantamagazine.org