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Nonuniqueness of site percolation on nonamenable quasi-transitive graphs

A mix of human-written and AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.CO — Combinatoricsmath.GR — Group Theorymath-ph — Mathematical Physics

Contributed by Yan Ru Pei ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Benjamini and Schramm conjectured in 1996 that site percolation on every nonamenable quasi-transitive graph has pc<pup_c<p_u. We prove this in Hutchcroft's operator form: for Bernoulli site percolation on every connected, locally finite, nonamenable quasi-transitive graph, the critical two-point operator is bounded on ℓ2\ell^2, pc<p2→2≤pup_c<p_{2\to2}\le p_u, and in the standard monotone coupling infinitely many infinite clusters coexist at every parameter of a nondegenerate interval. Since bond percolation is site percolation on the line graph, this contains OpenAI's recent bond theorem. For unimodular automorphism groups we adapt OpenAI's proof: cut vertices replace bridges, a pruning lemma handles retained contact vertices, a rejection step controls the junctions created by opening a vertex, and pivotal corridors become induced paths. For nonunimodular groups we extend Tran and Nguyen's site argument from vertex-transitive to quasi-transitive graphs.

Author comments: 60 pages: main text pp. 1-28, appendices A-G. An earlier version is on Zenodo, doi:10.5281/zenodo.24023209; this version differs only in adding the companion paper's DOI to its bibliography entry. The proof uses three unrefereed inputs, listed in the introduction: OpenAI's manuscript on nonuniqueness of bond percolation (both cases), Section 5 of Tran and Nguyen, arXiv:2609.07026 (nonunimodular case), and the companion paper "Critical site percolation dies on (almost) every quasi-transitive graph", doi:10.5281/zenodo.24023188 (critical finiteness and the nonunimodular case).

Provenance statement

Claude Opus 5.5 (Anthropic) was used for proof discovery and for writing the paper, running as agents in Claude Code. The contributor steered the work: choosing the problem (the site form of Benjamini-Schramm Conjecture 6, following OpenAI's bond theorem), directing the proof strategy, making the scope and attribution decisions, and reviewing the key mathematical lemmas. Independent model instances audited each component proof blind and rechecked every repair, and a model-run precedence search found no earlier statement of the site form. The proof relies on unrefereed inputs, listed in the introduction: OpenAI's bond-percolation manuscript, Section 5 of Tran and Nguyen (arXiv:2609.07026), and the companion paper (doi:10.5281/zenodo.24023188). No formal verification was done.

Tools used

Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1 · 60 pagesSubmitted by Yan Ru Pei60 pages: main text pp. 1-28, appendices A-G. An earlier version is on Zenodo, doi:10.5281/zenodo.24023209; this version differs only in adding the companion paper's DOI to its bibliography entry. The proof uses three unrefereed inputs, listed in the introduction: OpenAI's manuscript on nonuniqueness of bond percolation (both cases), Section 5 of Tran and Nguyen, arXiv:2609.07026 (nonunimodular case), and the companion paper "Critical site percolation dies on (almost) every quasi-transitive graph", doi:10.5281/zenodo.24023188 (critical finiteness and the nonunimodular case).CurrentNov 09, 2026