Nonuniqueness of percolation on nonamenable quasi-transitive graphs
We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's stronger operator-threshold conjecture, establishing $p_c\lt p_{2\to2}\le p_u$.
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The converse is true by work of Burton and Keane. I loved this problem so deeply. It said something so beautiful and simple about these highly treelike graphs which had somehow eluded proof in the general case. It’s why I fell in love with percolation theory. Unsurprisingly the OpenAI preprint…
whether there is a more natural proof for groups with free subgroups. At the same time, this problem deserved more endeavors. There were many ways ahead, and we could explore different routes to eventually reach the destination. It could be more humane. The journey could be enjoyed by people. But…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
for bond percolation on infinite connected locally finite quasi-transitive graphs with , extending the setting of my recent post on an AI proof of critical nonpercolation . It also announces for nonamenable graphs in this class , resolving the Benjamini–Schramm nonuniqueness conjecture discussed…