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\hypersetup{pdftitle={Nonuniqueness of site percolation on nonamenable quasi-transitive graphs},pdfauthor={Yan Ru Pei},pdfkeywords={site percolation, nonuniqueness, nonamenable graphs, quasi-transitive graphs, Benjamini--Schramm conjecture, operator norm}}
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\title{Nonuniqueness of site percolation\\on nonamenable quasi-transitive graphs}
\author{Yan Ru Pei\\[0.3em]\small\href{mailto:yanrpei@gmail.com}{\texttt{yanrpei@gmail.com}}}
\date{November 2026}

\begin{document}
\maketitle

\begin{abstract}
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.
\end{abstract}

\input{sections/intro}
\input{sections/prelim}
\input{sections/assembly}
\input{sections/goodness}
\input{sections/amplification}
\input{sections/critical}
\input{sections/nonunimodular}
\input{sections/discussion}

\appendix
\input{sections/app-tools}
\input{sections/app-goodness}
\input{sections/app-amplification}
\input{sections/app-critical}
\input{sections/app-cocycle}
\input{sections/app-b8}
\input{sections/app-table}

\input{sections/bibliography}

\end{document}