\section{Preliminaries}\label{sec:prelim}

\subsection{Conventions}\label{sec:conventions}

\paragraph{Reduction to a simple graph.} Site percolation on a graph with loops or multiple edges is
site percolation on its underlying simple graph $G_s$, which has the same vertex set and in which
$x\sim y$ if and only if $x\ne y$ are joined by at least one edge. The graph $G_s$ is infinite,
connected and locally finite; $\Aut(G)$ acts on $V$ through a subgroup of $\Aut(G_s)$, so $G_s$ is
quasi-transitive; and vertex boundaries, hence $h_V$, are unchanged. All site quantities
($p_c$, $p_u$, $\ptwo$, $T_p$ and the label coupling) coincide for $G$ and $G_s$. Unless stated
otherwise, \emph{graphs are simple from now on}, $d<\infty$ bounds the degrees, and unimodularity
refers to $\Aut(G_s)$. In contrast with~\cite[Lemma~2.1]{OAINU}, which subdivides multiple edges for
bond percolation, simplification adds no new sites.

\paragraph{Site notation.} A configuration is $\omega\in\{0,1\}^V$, with $1$ meaning open, and
$\Pp_p$ is the product Bernoulli($p$) law. Domains are vertex sets $D\subseteq V$ with their induced
graphs. The cluster $C_x^D$ is the set of vertices joined to $x$ by an open path inside $D$; it is
\emph{empty when $x$ is closed or $x\notin D$}. Put $C_x=C_x^V$, $s_x=|C_x|$ and
$T_p^D(x,y)=\Pp_p(y\in C_x^D)$, so $T_p(x,x)=p$. For $A\subseteq V$, $\partial A$ is the outer vertex
boundary, $\Nb[A]=A\cup\partial A$, $N(z)$ is the set of neighbours of $z$, and $\Nb[z]=\Nb[\{z\}]=
N(z)\cup\{z\}$. Inside a domain $D$, for $W\subseteq D$, put $\partial_DW=\partial W\cap D$ and
$\Nb_D[W]=W\cup\partial_DW$.
Every vertex of $\partial C_x$ is closed, and adjacent open vertices share a cluster. $\Adj$ is the
adjacency kernel. For a closed vertex $v$ put
\[
\widehat C_v=\{v\}\cup\bigcup_{a\sim v}C_a,
\]
the cluster of $v$ after opening it. If $C^{(1)},\dots,C^{(m)}$, $m\le d$, are the distinct open
clusters adjacent to a closed $v$, with sizes $n_i$, put $N_v=\sum_in_i=|\widehat C_v|-1$ and
$Q_v=\sum_{i\ne j}n_in_j$, a sum over ordered pairs.

We use the constants
\begin{equation}\label{eq:constants}
\kappa_p=\Bigl(\frac{1+d}{p}\Bigr)^2,\qquad C_{\mathrm{del}}(p)=\frac1p+d,\qquad
c_{\mathrm{nb}}(p)=1+\frac dp+\frac{d^2}{p^2},
\end{equation}
all bounded on $p\ge p_c/2$, and two consequences of Harris's inequality. (H1) If $z\sim a$ with
$a,z\in D$, then $p\,T_p^D(v,z)\le T_p^D(v,a)$, since $\{v\lr z\text{ in }D\}\cap\{a\text{ open}\}
\subseteq\{v\lr a\text{ in }D\}$; iterating, $p^lT_p^D(v,z)\le T_p^D(v,a)$ when $z$ and $a$ are
joined by a path of length $l$ in $D$. (H2) $\delta_{wy}\le T_p(w,y)/p$, since $T_p(y,y)=p$.
Consequently, entrywise,
\begin{equation}\label{eq:H}
I+\Adj\,T_p\le\frac{1+d}{p}\,T_p,\qquad I+T_p\Adj\le\frac{1+d}{p}\,T_p.
\end{equation}

Kernel facts from~\cite[Section~2]{OAINU}: if $|A(x,y)|\le B(x,y)$ with $B$ nonnegative and bounded,
then $\norm A\le\norm B$; if all row and column sums of $B\ge0$ are at most $b$, then $\norm B\le b$.

\subsection{Types, mass transport, trace and walk}\label{sec:types}

In the unimodular case we use the type averages of~\cite[Section~2.3]{OAINU}. Let
$\Gamma=\Aut(G)$ be unimodular, with vertex orbits $O_1,\dots,O_{h_0}$ and representatives
$o_1,\dots,o_{h_0}$. With a fixed Haar measure on $\Gamma$, put
\[
\sigma_i=\frac{\operatorname{vol}(\Gamma_{o_i})^{-1}}{\sum_j\operatorname{vol}(\Gamma_{o_j})^{-1}},
\qquad[F(o)]=\sum_{i=1}^{h_0}\sigma_iF(o_i),\qquad c_\sigma=(\min_i\sigma_i)^{-1}.
\]
A bracket $[F(o)]$ around a function of the root $o$ always denotes this finite type average; brackets
after $\E_p$ are ordinary expectation brackets. Operator norms, adjoints and positivity refer to
counting-measure $\ell^2(V)$. Let $P$ be the symmetric lazy walk, $P(x,y)=1/(2d)$ for $x\sim y$ and
$P(x,x)=1-\deg(x)/(2d)$. We use the following statements of the unrefereed preprint~\cite{OAINU} (Section~\ref{sec:earlier}) as
stated; they concern
only the graph, invariant operators, or finite product measures.

\begin{lemma}[{\cite[Lemma~2.7]{OAINU}}]\label{lem:MTP}
For every invariant nonnegative function $m$ on ordered vertex pairs,
$\bigl[\sum_ym(o,y)\bigr]=\bigl[\sum_xm(x,o)\bigr]$. The identity also applies to the expectation of
a covariant random mass and to absolutely summable signed or complex masses. In particular, an
invariant stochastic kernel on a single orbit has column sums one.
\end{lemma}

\begin{lemma}[{\cite[Lemma~2.8]{OAINU}}]\label{lem:trace}
For a bounded invariant operator $S$, $\tr S=[S(o,o)]$ is a positive normalized trace: $\tr I=1$ and
$\tr(AB)=\tr(BA)$ for bounded invariant $A,B$. If $A$ is positive semidefinite, invariant and
bounded, and $B=B^*$ is invariant and bounded, then $|\tr(BA)|\le\norm B\tr A$ and
$A(x,x)\le c_\sigma\tr A$.
\end{lemma}

\begin{lemma}[{\cite[Lemma~2.9]{OAINU}}]\label{lem:walk}
$P$ is positive semidefinite and $\rho:=\norm P<1$. Its projection onto the vertex types is an
irreducible lazy finite Markov chain, and there are $c_*>0$ and $l_0<\infty$ with
$\sum_{y\in O_j}P^l(x,y)\ge c_*$ for every $x\in V$, $j\le h_0$ and $l\ge l_0$.
\end{lemma}

The bound $\rho<1$ is where nonamenability enters the unimodular argument; Section~\ref{sec:discussion}
lists its uses.

\subsection{Site tools}\label{sec:tools}

The following site forms of the preliminary lemmas of~\cite[Section~2]{OAINU} are proved in
Appendix~\ref{app:tools}. They are routine adaptations, except that the Russo identity acquires a
remainder term. Lemma~\ref{lem:crit-finite} holds on every graph of Theorem~\ref{thm:main}; the
others are stated where they are used. Put $\chi_x(p)=\E_ps_x$ and $\chi_{\max}(p)=\sup_x\chi_x(p)$,
and, when $\Aut(G)$ is unimodular, $\chi(p)=[\chi_o(p)]$.

\begin{lemma}[Critical finiteness]\label{lem:crit-finite}
Let $G$ be as in Theorem~\ref{thm:main}, simple. Then $0<p_c<1$, and at $p_c$ all open clusters are
finite almost surely, simultaneously at every vertex and in every induced subgraph under restriction
of the full configuration.
\end{lemma}

The lower bound is path counting, the upper bound a Peierls argument with the closed outer vertex
boundary, and finiteness at $p_c$ is~\cite[Theorem~1.1(2)]{Pei26b}, since $h_V>0$ forces exponential
growth.

\begin{lemma}[Association, disjoint witnesses, tree bound]\label{lem:tools}
Increasing nonnegative functions of the site bits are positively correlated, and increasing events
with disjoint finite open witnesses satisfy the van den Berg--Kesten inequality. For all $x,y,z$ and
every domain $D$,
\begin{equation}\label{eq:tree}
\Pp_p(x\lr y,\ x\lr z\text{ in }D)\le\kappa_p\sum_wT_p^D(x,w)T_p^D(w,y)T_p^D(w,z).
\end{equation}
In particular $\E_ps_x^2\le\kappa_p\chi_{\max}(p)^3$.
\end{lemma}

\begin{lemma}[{Threshold order; site form of~\cite[Lemma~2.4]{OAINU}}]\label{lem:threshold-order}
For every infinite, connected, locally finite, quasi-transitive graph, $\ptwo\le p_u$ for site
percolation.
\end{lemma}

\begin{lemma}[{Subcritical finiteness and type comparison; site form of~\cite[Lemma~2.10]{OAINU}}]
\label{lem:sharpness}
Let $\Aut(G)$ be unimodular. For $p<p_c$: $\chi_{\max}(p)<\infty$, $\sup_x\E_ps_x^2\le\kappa_p
\chi_{\max}(p)^3$ and $\chi_{\max}\le c_\sigma\chi$. There is $c>0$ with $\chi_x(p)\ge c\chi(p)$ for all
$x$ and $p_c/2\le p<p_c$. The kernel $T_p$ is symmetric, positive semidefinite and bounded, with
$\norm{T_p}\le\chi_{\max}(p)$.
\end{lemma}

Finiteness of $\chi_{\max}$ below $p_c$ is site sharpness on quasi-transitive
graphs~\cite[Theorem~2]{AV08}, where~\cite[Lemma~2.10]{OAINU} uses the bond argument of~\cite{DT16};
the other assertions are proved in Appendix~\ref{app:tools}.

\begin{lemma}[{Russo identities in operator norm; site form of~\cite[Lemma~2.11]{OAINU}}]
\label{lem:russo}
Let $\Aut(G)$ be unimodular. The map $p\mapsto T_p$ is locally absolutely continuous in operator norm
on $(0,p_c)$, and so is $\chi$. For almost every $p\in(0,p_c)$,
\begin{equation}\label{eq:russo-a}
(1-p)\,T_p'(x,y)=\sum_v\Pp_p\bigl(v\text{ closed},\ x\in\widehat C_v,\ y\in\widehat C_v,\ x\nlr y\bigr),
\end{equation}
equivalently $(1-p)T_p'=K^c_p+R_p$ with the nonnegative symmetric invariant kernels
\[
K^c_p(x,y)=\E_p\bigl[k_s(C_x,C_y);\ x,y\text{ open},\ C_x\ne C_y\bigr],\qquad k_s(A,B)=|\partial A\cap
\partial B|,
\]
$R_p(x,y)=\Pp_p(x\text{ closed},\ y\in\widehat C_x)+\Pp_p(y\text{ closed},\ x\in\widehat C_y)$ for
$x\ne y$, and $R_p(x,x)=1-p$. Moreover, entrywise,
\begin{equation}\label{eq:russo-b}
0\le R_p\le(1-p)I+\Adj\,T_p+T_p\Adj,\qquad(1-p)T_p'\le(I+T_p\Adj)(I+\Adj\,T_p),
\end{equation}
and
\begin{equation}\label{eq:russo-c}
(1-p)\chi'=\bigl[\E_p\1\{o\text{ closed}\}(1+2N_o+Q_o)\bigr].
\end{equation}
With $\Psi(p)=\chi^{-2}\bigl[\E_p\1\{o\text{ closed}\}Q_o\bigr]$,
\begin{equation}\label{eq:russo-d}
(1-p)\chi'\ge\chi^2\Psi(p),\qquad(1-p)\chi'\le\chi^2\Psi(p)+1+2d\chi_{\max}.
\end{equation}
\end{lemma}

The pointwise identity~\eqref{eq:russo-a} is the site Russo formula combined with the identity
\[
\Pp(v\text{ pivotal})=(1-p)^{-1}\Pp(v\text{ closed and pivotal}),
\]
as in the proof
of~\cite[Proposition~19]{AV08} and in~\cite[Auxiliary Lemma~5.A]{TN26}. The new feature is the
remainder $R_p$, from the pivotal sites $v\in\{x,y\}$. In~\cite{OAINU} the Russo identity enters only
the proof of Proposition~\ref{prop:moments}, which needs the lower bound in~\eqref{eq:russo-d} and local
absolute continuity, and the proof of Proposition~\ref{prop:trace}, which
needs~\eqref{eq:russo-a}--\eqref{eq:russo-b}; there the remainder contributes at most a constant
multiple of the triangle trace (Section~\ref{sec:moments}). In~\cite{OAINU}, $D(p)$ denotes the bond
analogue of $\Psi(p)$.

\begin{lemma}[{Sprinkling; site form of~\cite[Lemma~2.12]{OAINU}}]\label{lem:sprinkling}
For $0\le p,a\le1$, entrywise,
\begin{equation}\label{eq:sprinkling}
T_{p+(1-p)a}\le T_p+\sum_{j\ge1}a^j(I+T_p\Adj)\bigl[\Adj(I+T_p\Adj)\bigr]^{j-1}(I+\Adj\,T_p).
\end{equation}
If $T_p$ is bounded and $ad(1+d\norm{T_p})<1$, the left side is bounded. In particular, if $p_c<1$,
boundedness of $T_{p_c}$ implies $p_c<\ptwo$.
\end{lemma}

\begin{lemma}[{Monotone passage; site form of~\cite[Lemma~9.2]{OAINU}}]\label{lem:passage}
Let $G$ be an infinite locally finite graph, let $p_*\in(0,1]$, and let $T_p$ be the Bernoulli site
two-point kernel of $G$ at parameter $p$. Then $T_{p_*}$ is bounded on counting-measure $\ell^2(V)$ if and only if
$\sup_{0\le p<p_*}\norm{T_p}_{2\to2}<\infty$. In that case the supremum equals $\norm{T_{p_*}}_{2\to2}$.
\end{lemma}

\begin{proposition}[{One coupling for an interval; site form of~\cite[Proposition~9.3]{OAINU}}]
\label{prop:coupled}
Let $G$ be infinite, connected, locally finite and quasi-transitive. Suppose that
$p_c<p_1<p_2<1$ and that $T_{p_2}$ is bounded on $\ell^2(V)$. In the coupling by independent uniform
labels on the vertices, almost surely there are infinitely many infinite clusters simultaneously at
every $p\in[p_1,p_2]$.
\end{proposition}

As noted in Section~\ref{sec:earlier}, Proposition~\ref{prop:coupled} also follows
from~\cite[Theorem~1.3]{HPS99} with its remark on site percolation, once $p_c<p_u$ is known.