The continuity of the percolation transition remained an open question between dimensions 3 and 10. Formal proof provided by Anthropic’s models establishes this, leaving mathematicians divided.
Born in 1957 from the work of Simon Broadbent and John Hammersley on the flow of a fluid into a porous medium, percolation theory studies the moment when random bonds in a network eventually form an infinite cluster. In 1980, Harry Kesten determined that the critical threshold of the square plan network was 1/2. Was a central question: at the critical point, is there already an infinite cluster, or is the transition continuous? The answer was known in dimension 2 and beyond 10, but dimensions 3 to 10 have resisted for decades.
Formal evidence, mixed feelings
Anthropic has published a formal demonstration, written by his Claude models, that the transition is continuous in all these dimensions — in other words, that no infinite clusters appear at the critical threshold. The result came shortly after Hugo Duminil-Copin, Fields 2022 medal, predicted that an AI would solve this problem he called Grail.
The Scientific American goes back to the community's reactions. Probabilist Benedikt Jahnel describes an ambivalent feeling: the joy of seeing the conjecture finally cut, and a form of disenchantment to the idea that the decisive step came from a machine. Duminil-Copin believes that a human would probably have won a Fields medal for such a result, which raises the question of the future of the great mathematical distinctions.
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