\section{Introduction}\label{sec:intro}

\subsection{The site form of the Benjamini--Schramm conjecture}

Let $G=(V,E)$ be an infinite, connected, locally finite graph. In \emph{Bernoulli site percolation}
with parameter $p\in[0,1]$, each vertex is \emph{open} independently with probability $p$ and
\emph{closed} otherwise; $\Pp_p$ denotes the law. The open cluster $C_x$ of a vertex $x$ is the set
of vertices joined to $x$ by a path all of whose vertices, endpoints included, are open; in particular
$C_x=\varnothing$ when $x$ is closed. Write $x\lr y$ when $y\in C_x$, and let $N_\infty$ be the
number of infinite open clusters. The two thresholds of the phase diagram are
\[
p_c(G)=\inf\{p\in[0,1]:\Pp_p(N_\infty\ge1)>0\},\qquad
p_u(G)=\inf\{p\in[0,1]:\Pp_p(N_\infty=1)=1\}.
\]
The graph is \emph{quasi-transitive} if $\Aut(G)$ has finitely many orbits on $V$, and then its
degrees are bounded; $d$ denotes the maximal degree. For finite nonempty $A\subset V$ let
$\partial A=\{v\in V\setminus A:v\sim a\text{ for some }a\in A\}$ be the outer vertex boundary, and put
\[
h_V(G)=\inf_{0<|A|<\infty}\frac{|\partial A|}{|A|}.
\]
The graph is \emph{nonamenable} if $h_V(G)>0$; for bounded degree this is equivalent to positivity of
the edge isoperimetric constant.

Benjamini and Schramm's paper on percolation beyond $\Z^d$ is written for site percolation:
``Throughout the paper, site percolation is discussed. Except when dealing with planar graphs, the
results and questions remain equally valid, with only minor modifications, for bond percolation''
\cite[p.~72]{BS96}. Their Cheeger constant is $h_V$, defined with the vertex boundary
\cite[p.~73]{BS96}, and their nonuniqueness conjecture reads
\begin{quote}
``Conjecture 6 Assume that the almost transitive graph $G$ has positive Cheeger constant. Then
$p_c(G)<p_u(G)$.'' \cite[p.~76]{BS96}
\end{quote}
where almost transitive means quasi-transitive \cite[p.~72]{BS96}. On amenable quasi-transitive
graphs the infinite cluster is unique whenever it exists, by the argument of Burton and
Keane~\cite{BK89}, as noted in~\cite[p.~76]{BS96}, so $p_c=p_u$ there; the conjecture asserts that
nonamenability alone produces a nonuniqueness phase.

We study the stronger operator form of Hutchcroft~\cite[Proposition~2.3 and Conjecture~6.1]{Hut19}. Let
$T_p(x,y)=\Pp_p(x\lr y)$ be the two-point kernel; for site percolation $T_p(x,x)=p$. Equip $V$ with
counting measure, let $T_p$ act on finitely supported functions by $(T_pf)(x)=\sum_yT_p(x,y)f(y)$,
and put
\[
\ptwo(G)=\sup\{p\in[0,1]:T_p\text{ extends to a bounded operator on }\ell^2(V)\}.
\]
For bond percolation the conjecture, together with the strict operator gap $p_c<\ptwo$, was recently
proved on every nonamenable quasi-transitive graph in a manuscript released by
OpenAI~\cite{OAINU}. That theorem does not give the site form.
The site form is the more general one: bond percolation on a graph $H$ is site percolation on its line
graph, which is quasi-transitive and nonamenable when $H$ is (Appendix~\ref{app:bond}), so a site
theorem implies the corresponding bond theorem; no reduction in the other direction is available.

\subsection{Results}

\begin{theorem}\label{thm:main}
Let $G$ be an infinite, connected, locally finite, quasi-transitive graph with $h_V(G)>0$. For
Bernoulli site percolation on $G$:
\begin{enumerate}[label=\textup{(\arabic*)},leftmargin=2.2em]
\item $\sup_{0\le p<p_c}\norm{T_p}_{2\to2}=\norm{T_{p_c}}_{2\to2}<\infty$;
\item $p_c(G)<\ptwo(G)\le p_u(G)$;
\item there are $p_c<p_1<p_2<1$ such that, in the coupling by independent uniform vertex labels,
almost surely there are infinitely many infinite clusters simultaneously for every $p\in[p_1,p_2]$.
\end{enumerate}
\end{theorem}

Loops and multiple edges are allowed: site percolation depends only on adjacency
(Section~\ref{sec:conventions}). In the coupling of~(3), each vertex $v$ carries an independent
uniform label $U_v\in[0,1]$, and $v$ is open at parameter $p$ when $U_v\le p$.

\begin{corollary}\label{cor:bond}
Let $G$ be an infinite, connected, locally finite, quasi-transitive graph with $h_V(G)>0$. Then the
conclusions \textup{(1)--(3)} of Theorem~\ref{thm:main} hold for Bernoulli bond percolation on $G$,
with uniform labels on the edges in~\textup{(3)}. That is, Theorem~\ref{thm:main} contains the bond
theorem~\cite[Theorem~1.1]{OAINU}.
\end{corollary}

Corollary~\ref{cor:bond} follows from Theorem~\ref{thm:main} applied to the line graph $L(G)$. With $d$
the maximal number of edges at a vertex of $G$, the graph $L(G)$ is quasi-transitive with $h_V(L(G))\ge
h_V(G)/d$; the open site clusters of $L(G)$ are the edge sets of the bond clusters of $G$ with at
least one edge; and, with the vertex--edge incidence
kernel $B$, $T^{\mathrm{bond}}_p\le I+B\,T^{\mathrm{site},L(G)}_pB^*$ entrywise and
$\norm B^2\le2d$. The proof is in Appendix~\ref{app:bond}.

\subsection{Relation to earlier work}\label{sec:earlier}

\paragraph{Site percolation.} The site inequality $p_c<p_u$ was known under hypotheses beyond
nonamenability. Benjamini and Schramm proved $0<p_c<p_u<1$ for site and bond percolation on
one-ended nonamenable transitive planar graphs~\cite[Theorem~1.1]{BS01}, after Lalley's work on site
percolation on Fuchsian groups~\cite{Lal98}. Schonmann's theorem~\cite{Sch01} gives
$p_c^{\mathrm{site}}<p_u^{\mathrm{site}}$ on transitive graphs of degree $d$ with
$h_V(G)/d\ge1/\sqrt2$~\cite[Corollary~7.39(ii)]{LP16}; the vertex-isoperimetric constant $\Phi_V$
of~\cite[p.~180]{LP16}, with unit weights, is $h_V$. Czajkowski proved a
nonuniqueness phase for bond and site percolation on reflection groups of some polyhedra in
$\mathbb H^3$~\cite{Cza13}. Li proved site nonuniqueness for properly embedded planar graphs of minimum
degree at least seven~\cite{Li23} and under an end-structure hypothesis for planar graphs~\cite{Li26}.
For vertex-transitive graphs with nonunimodular automorphism group, Tran and Nguyen proved the site
tilted-threshold separation $p_c<p_c(\lambda)$~\cite[Proposition~5.5]{TN26}, where $p_c(\lambda)$ is the
supremum of the parameters at which the tilted susceptibility $\sum_xT_p(u,x)\Delta(u,x)^\lambda$, with
$\Delta$ the modular function, is bounded in $u$ (Section~\ref{sec:assembly}). Together with Hutchcroft's operator
argument this gives the conclusions (1)--(2) of Theorem~\ref{thm:main} for those graphs
(Proposition~\ref{prop:vt}). We are not aware of an earlier statement of $p_c<p_u$ for site percolation
on all nonamenable quasi-transitive graphs.

\paragraph{The bond theorem.} The unimodular case of Theorem~\ref{thm:main} follows the architecture
of~\cite{OAINU}, described in Section~\ref{sec:ideas}. Every result whose proof changes for sites is
stated here and each changed step is proved; steps that carry over are cited by their number
in~\cite{OAINU}, and Appendix~\ref{app:table} lists the correspondence. In the nonunimodular
case,~\cite{OAINU} uses Hutchcroft's bond theorem $p_c<p_{q\to q}$~\cite[Theorem~2.9]{Hut19}, where
$p_{q\to q}$ is the $\ell^q$ analogue of $\ptwo$; it rests on the bond tilted-susceptibility theory
of~\cite{HutNU}. For sites we replace it by a quasi-transitive form of Tran and Nguyen's site
argument~\cite[Section~5]{TN26}.

\paragraph{Known components.} Three of the conclusions are known or routine.
The inequality $\ptwo\le p_u$ holds because uniqueness forces $T_p(x,y)\ge\theta^2$ by Harris's
inequality~\cite[Lemma~2.4]{OAINU}, \cite[p.~269]{LP16}, \cite[Proposition~1.3]{Li23}. The coupled
interval is a consequence of the simultaneous phase theorem of H\"aggstr\"om, Peres and
Schonmann~\cite[Theorem~1.3]{HPS99}, whose results are stated for bond percolation with the remark
``All our results and proofs may be adapted to site percolation as well''~\cite[Section~1]{HPS99};
we include the direct proof of~\cite[Proposition~9.3]{OAINU} for sites
(Proposition~\ref{prop:coupled}). The supremum identity in~(1) is monotone convergence. Critical
finiteness, the absence of an infinite cluster at $p_c$, is the site theorem of the companion
paper~\cite[Theorem~1.1(2)]{Pei26b}, whose nonunimodular part~\cite[Theorem~1.2]{Pei26b} is also
used in the nonunimodular case (Appendix~\ref{app:b8}).

\paragraph{Unrefereed inputs.} The nonunimodular case of Theorem~\ref{thm:main} depends on Tran and
Nguyen's arXiv preprint~\cite{TN26}, posted in September 2026 and not refereed. We reuse the arguments
of its Section~5, change the steps listed in Appendix~\ref{app:b8} and prove each replacement, and use
the other steps as written. The unimodular case does not use~\cite{TN26}. Both cases use~\cite{OAINU},
which is also unrefereed, and the companion paper~\cite{Pei26b}, also a preprint.

\subsection{Ideas of the proof}\label{sec:ideas}

The theorem reduces to bounding $T_{p_c}$: sprinkling then gives $p_c<\ptwo$, and the remaining
conclusions follow from the known components above (Section~\ref{sec:assembly}). After the passage to
the underlying simple graph, we split according to whether $\Aut(G)$ is unimodular.

\paragraph{The unimodular case: the architecture of~\cite{OAINU}.} We first describe the bond argument in
its own terms. Fix a cap $M$; the open cluster of a vertex $x$ is \emph{capped} if it has at most $M$
vertices. In a connected piece of a cluster, the \emph{pivotal distance} between two vertices is the
number of bridges separating them, and a capped cluster is \emph{good} if, whenever a set of its vertices
is deleted, every remaining piece has pivotal diameter at most $KM^\tau$, where $K$ is the number of open
edges from the piece to the deleted set and $\tau$ is slightly larger than $1/2$. A \emph{local endpoint map}
is a randomized rule that reads the configuration near $x$ and either fails or outputs a vertex $y$ of
the capped good cluster of $x$ in a fixed orbit; it is measured by its success probability $u$ and by
the norm $r$ of its \emph{endpoint kernel} $K(x,y)=\Pp(\text{output }y\mid\text{success at }x)$. Suppose
that the critical moment $\E_{p_c}|C_o|^\alpha$ at a vertex $o$, averaged over the finitely many vertex
types, is infinite for some $0<\alpha<1/2$. Then an \emph{amplification} combines $k$ independent copies
of a map into a new one. A chain of $k$ old steps runs from a vertex $z$ into the cluster $A$ of the root
$x$, and the \emph{switching} splices the chain into $A$ step by step: each step takes the piece of its
own cluster, in its own independent configuration, that hangs off the current root set, and joins it to
that set through a contact edge, the \emph{seam}, by exchanging the bits of the two configurations on a
mask. The \emph{inverse} of the switching locates the seams among the separating bridges of the output
cluster by a code of gaps, which goodness keeps short, and undoes the exchanges; a second-moment bound
undoes two inverses at once. Iterating the amplification produces maps whose endpoint kernels become
dispersive much faster than their success probability decays or their cap grows. Run just below $p_c$
in a \emph{deleted domain}, the complement of an independent cluster, such a map contradicts the
divergent moment. The resulting fractional moments give an almost linear susceptibility bound and a
subpower bound for the triangle trace, and a deformation of $T_p$, which penalizes connections forced
through a \emph{corridor} of pivotal edges, the loop erasure of a random-walk path, shows that $T_p$ cannot blow up at
$p_c$.

\paragraph{What changes for sites.} Each bond-specific object changes for sites, and five changes carry
new ideas.
\begin{enumerate}[label=\textup{(\roman*)},leftmargin=2.2em]
\item \emph{Cut vertices in place of bridges.} For sites an open edge cannot be closed independently of
its endpoints, so the separating objects are the cut vertices of a piece, its pivotal sites, and
pivotal distance counts separating cut vertices. Its triangle inequality carries an extra $+1$, which
costs only constants (Section~\ref{sec:goodness}).

\item \emph{Pruning a retained contact.} To prove goodness,~\cite{OAINU} closes all but one contact
edge of a bad piece $B$, so that $B$ hangs from the rest of the cluster by a single edge and the
pivotal distances inside $B$ persist in the modified cluster. For sites the retained object is a contact vertex $a$, which
may have several neighbours in $B$; if $B$ is a path and $a$ is adjacent to both ends, $B\cup\{a\}$ is
a cycle with no cut vertices at all. We also close all but one neighbour $b^*$ of $a$ in $B$, and a
Voronoi-cell argument chooses $b^*$ so that the pivotal distance drops by at most a factor $2d$
(Lemma~\ref{lem:pruning}).

\item \emph{Junction rejection.} For bonds, opening a contact edge joins exactly two clusters. For
sites the seam is a closed vertex on the boundary of the root set, and opening it attaches every open
component adjacent to it, up to $d-1$ foreign branches. We reject the splice unless every neighbour of
the seam outside the closed neighbourhoods of the current root set and of the inserted piece is closed
(Section~\ref{sec:switching}). These rejection bits have not been read before, so each step costs at
most a factor $(1-p_c)^{d-1}$; this cost is linear in the number of steps in the exponent and is
absorbed by the amplification, while the inverse of the switching stays unambiguous.

\item \emph{Joint restoration in the second moment.} The second-moment bound of~\cite{OAINU} undoes two
switchings whose inserted pieces are far apart, and uses that removing both inserted parts, the sets
hanging at the two first seams, leaves a connected set containing the root. For bonds the two first
seams are bridges, and this holds. For sites they are cut vertices, which can lie on one cycle through
the root, and removing both hanging sets can disconnect the root's set. We record the two seams and
their neighbourhoods in the remaining set, at a polynomial cost, and replay the switching from a
possibly disconnected set (Lemma~\ref{lem:second-moment}). The root cluster of the remaining set can then
be much smaller than that set, so a lower size window used in~\cite{OAINU} is lost; a second-moment bound at the
scales selected in Lemma~\ref{lem:scales} replaces it.

\item \emph{Induced corridors.} The deformation penalizes connections for which every vertex of a
prescribed corridor is pivotal. For sites a corridor must be an induced path with closed blockers
around it, since a chord or an open neighbour would bypass its pivotal vertices. We take the greedy
shortcut of the loop erasure of the random-walk path, which is induced, and its interior is entered and
left only through its two endpoints (Lemma~\ref{lem:isolation}). When a connection through a corridor is
split into two legs at the corridor, the corridors whose interior meets the first one or its endpoints
form a new class, of bounded total weight (Lemma~\ref{lem:legs}).
\end{enumerate}

The other changes are local: in the site Russo identity in operator norm, the derivative of $T_p$ is a
sum over closed pivotal sites, which splits into a contact kernel $|\partial C_x\cap\partial C_y|$ and a
remainder bounded by $(1-p)I+\Adj\,T_p+T_p\Adj$ (Lemma~\ref{lem:russo}); deleted domains become
$V\setminus\Nb[A]$ with $\Nb[A]=A\cup\partial A$; and several constants depend on the degree.

\paragraph{The nonunimodular case.} Hutchcroft's operator argument bounds $\norm{T_p}$ by the tilted
susceptibility $\sup_u\sum_xT_p(u,x)\Delta(u,x)^{1/2}$; the argument is model-free
(Section~\ref{sec:cocycle-main}). It therefore suffices to show that the site tilted susceptibility is
finite slightly above $p_c$. Tran and Nguyen proved this for vertex-transitive
graphs~\cite[Proposition~5.5]{TN26}. Their argument uses transitivity in many places: neighbours one
layer up or down in height, where the height of $z$ relative to $v$ is $\log\Delta(v,z)$; constants
uniform in the root; and transports at a fixed root. On a quasi-transitive graph such neighbours need
not exist, for instance at pendant vertices, and lower bounds uniform in the root fail at
\emph{plateau} vertices, whose neighbours all have the same height. We replace these uses by monotone
paths with length bounds, connectors that end in a prescribed orbit, a tilted mass transport averaged
over a random root, comparisons between orbits, and a transfer of peak estimates to a \emph{sloped}
vertex (Section~\ref{sec:b8-devices}). The decisive step is the quasi-transitive form of the surgery
inequality~\cite[(5.76)]{TN26}, the site analogue of~\cite[Lemma~6.4]{HutNU}, which we apply at a sloped
vertex (Proposition~\ref{prop:DT}).

\subsection{Organization}

Section~\ref{sec:prelim} fixes conventions and states the site tools, whose proofs are in
Appendix~\ref{app:tools}. Section~\ref{sec:assembly} proves Theorem~\ref{thm:main} from two case
theorems, Theorem~\ref{thm:unimod} (unimodular) and Theorem~\ref{thm:b8} (nonunimodular); the proof of
Corollary~\ref{cor:bond} is in Appendix~\ref{app:bond}. Sections~\ref{sec:goodness}--\ref{sec:critical}
prove Theorem~\ref{thm:unimod}: good critical clusters with the pruning lemma, the amplification with
junction rejection and joint restoration in the second moment, and the critical bound with induced
corridors. The routine site adaptations of~\cite{OAINU} used there are proved in
Appendices~\ref{app:goodness}, \ref{app:amplification} and~\ref{app:critical}.
Section~\ref{sec:nonunimod} proves Theorem~\ref{thm:b8}; the vertex-transitive deduction is in
Appendix~\ref{app:cocycle}, and the proofs of the devices and the change list against~\cite[Section~5]{TN26}
are in Appendix~\ref{app:b8}. Appendix~\ref{app:table} lists the correspondence with the numbered
statements of~\cite{OAINU}. Section~\ref{sec:discussion} records where the hypotheses enter.