The Benjamini–Schramm nonuniqueness conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The Benjamini–Schramm nonuniqueness conjecture

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The Benjamini–Schramm nonuniqueness conjecture. Proves $p_c\lt p_u$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.

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LEVEL 1 Nonuniqueness of percolation on nonamenable quasi-transitive graphs

released 2026-09-24 | 2 theorems · 32 lemmas · 46 proofs · 23,841 words | PLAY LEVEL 1 » (pdf)

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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