Yan Ru Pei

math.PR — Probability

Nonuniqueness of site percolation on nonamenable quasi-transitive graphs

Contributed by Yan Ru Pei

Benjamini and Schramm conjectured in 1996 that site percolation on every nonamenable quasi-transitive graph has $p_c<p_u$. 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 $\ell^2$, $p_c<p_{2\to2}\le p_u$, 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).