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 dimensions.

In 2024 Gady Kozma and Shahaf Nitzan showed how to derive the dying percolation conjecture from a proposed conjecture about percolation on general graphs. (I briefly discussed it in this post. Their conjecture was so general that many of us expected a counterexample to be discovered before long.) The dying percolation conjecture asserts that for percolation in $\mathbb Z^d$ at the critical probability, with probability one, there is no infinite cluster. This was known for planar percolation and for percolation in high dimensions. It was a famous open problem in the intermediate dimensions starting with dimension 3. A Claude document, accompanied with a Lean verification, claims a 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 about three-dimensional percolation. Is critical percolation in dimension three noise sensitive? Is the total influence of critical finite percolation (for large finite n by n by n box) larger than $\log^{1+\epsilon} n$ ? Larger than some $n^{\alpha}$ ? (Here $\epsilon, \alpha$ are positive.)
(Update: Gady Kozma reminded me that the dying percolation conjecture is stronger than noise sensitivity. So critical percolation is noise sensitive.)

I had a long research project around these questions with Gady, in which we managed to prove some interesting lemmas. We hoped to bring influences and Fourier tools to the picture.

Hugo’s Duminil-Copin’s post on “Proofs and Prompts”

There is a very interesting new blog called Proofs and Prompts and Hugo Duminil-Copin wrote a thoughtful post Care for a little more AI? about AI and mathematics, using the θ(p꜀) = 0 problem as a primary example. This was three days before Claude claimed the proof and several commentators remarked about the new AI proof.

I liked Alonso Castillo-Ramirez’ comment: “we can still have great joy and be marvelled by the beauty of mathematics on its own, independently if it was created by a human or an AI.”

And here is a moving Facebook post by another famous researcher in percolation theory – Jeff Steif.

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