%% ================================================================
\section{Review of the two-dimensional case}

Our goal is to extend the results of section VI.C of \cite{bib:pcalw} from two
dimensions to three.  We begin by summarizing the two-dimensional situation.

xxx

Let
$$
	\pi = P[A \in \Cinf \cup A' \in \Cinf] = P[B \in \Cinf \cup B' \in \Cinf].
$$
We want an upper bound on $\pi^2$.  xxx compare to other.

xxx $\theta$.

Using the inclusion-exclusion principle, we have
\begin{align*}
	P[A \in \Cinf \cup A' \in \Cinf] &= P[A \in \Cinf]
		+ P[A' \in \Cinf] - P[A' \in \Cinf \cap A' \in \Cinf].
\end{align*}
The first two terms are both $\theta$.  Using the transitivity of the
clustering relation we may rewrite the last term as well.  One obtains
\begin{align*}
	P[A \in \Cinf \cup A' \in \Cinf]
		&= 2\theta - P[A \in \Cinf \cap A \oo A'].
\end{align*}
We now desire a lower bound on the last term.
Using the FKG inequality \cite{bib:grimmett} [xxx quack about increasing events],
\begin{align*}
	P[A \in \Cinf \cap A \oo A']  \ge
	P[A \in \Cinf] P[A \oo A'].
\end{align*}
Now, $P[A \in \Cinf]$ is simply $\theta$; write
$$
	\tau = P[A \oo A'].
$$
Then
$$
	P[A \in \Cinf \cap A \oo A']  \ge
	\theta\tau.
$$
For our upper bound on $\pi$ we now have
$$
	\pi \le \theta(2-\tau).
$$


%% ================================================================
\section{The three-dimensional case}

%% ----------------------------------------------------------------
\subsection{Lattice doubling}

xxx picture here

%% ----------------------------------------------------------------
\subsection{Upper bounds}

%% ----------------------------------------------------------------
\subsection{Monte Carlo estimation of connectivity functions}

%% ----------------------------------------------------------------
\subsection{Curve fitting}

%% ================================================================
\section{Conclusion}

\begin{vim_bug_workaround}
\end{vim_bug_workaround}
