Amazing: There is no Percolation at the Critical Probability in all Dimensions. (Solved by AI via a conjecture of Gady Kozma and Shahaf Nitzan.)

The θ(p꜀) = 0 conjecture is solved in all dimension. In 2024 Gady Kozma and Shahaf Nitzan showed a derivation of the dying percolation conjecture from a proposed conjecture about percolation for general graphs. (I briefly discussed it in this post. Their conjecture was so general that many of us expected a counterexample to be discovered not before long.) The dying percolation conjecture asserts that for percolation in at the critical probability, with probability one, there is no infinite cluster. This was known for planar percolation and for percolation in high dimension. It was a famous open problem in the intermediate dimensions starting with dimension 3. A Claude document accompanied with a Lean verification claims positive solution to the Kozma-Nitzan conjecture. (h/t to Itai Benjamini who told me about it yesterday and also about Hugo’s post.) If verified, this is a remarkable breakthrough. See here and here for the AI’s documents. There are very interesting related question…