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/Author (John Kerl)
/Title  (Lattice-quadrupling notes)
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\begin{document}
\title{Lattice-quadrupling notes}
\author{John Kerl}
\date{\today}

\maketitle

%% ================================================================
\section{Correlation length}

Abstract definition of correlation length $\xi(p)$ for bond percolation,
where $C_0$ is the cluster containing the origin:
\begin{align*}
	\xi(p) &= \frac{
		\sum_x |x|\; P(x \in C_0)
		}{
		\sum_x P(x \in C_0)
		} \\
%
	\textrm{or} \quad
	\xi(p) &= \left[ \frac{
		\sum_x |x|^2\; P(x \in C_0)
		}{
		\sum_x P(x \in C_0)
		} \right]^{1/2}
		\\
%
	\textrm{i.e. in general} \quad
	\xi(p) &= \left[ \frac{
		\sum_x |x|^q\; P(x \in C_0)
		}{
		\sum_x P(x \in C_0)
		} \right]^{1/q}.
\end{align*}

Computational issue:  As is, the sum over lattice sites $x$ is on the outside.
The probabilities, to be estimated by sample means taken over $R$ realizations,
are on the inside.  Na\"ively following this recipe, we would generate $R$
realizations for each of the $L^d$ lattice sites $x$.  It is far more efficient
to loop over all $x$, inside a loop over realizations.  Thus for each single
realization, we can estimate $P(x \in C_0)$ for all $x$ in the lattice.

Recall that $P(A) = \bbE[1_A]$.  Then, with $\bbE$ being the expectation
over all realizations, estimated by sample mean over $R$ realizations,
we have
\begin{align*}
	\xi^q(p) &= \frac{
		\sum_x |x|^q\; P(x \in C_0)
		}{
		\sum_x \qquad      P(x \in C_0)
		} \\
%
	&= \frac{
		\sum_x |x|^q\; \bbE[ 1(x \in C_0) ]
		}{
		\sum_x \qquad  \bbE[(x \in C_0)]
		} \\
%
	&= \frac{
		\bbE\left[ \sum_x |x|^q\; 1(x \in C_0) \right]
		}{
		\bbE\left[ \sum_x \qquad  1(x \in C_0) \right]
		} \\
%
	&\approx \frac{
		1/R \sum_{r=1}^R \sum_x |x|^q\; 1(x \in C_0)
		}{
		1/R \sum_{r=1}^R \sum_x \qquad  1(x \in C_0)
		} \\
%
	&= \frac{
		\sum_{r=1}^R \sum_x |x|^q\; 1(x \in C_0)
		}{
		\sum_{r=1}^R \sum_x \qquad  1(x \in C_0)
		}.
\end{align*}

This is for $p<p_c$.  For $p>p_c$, replace
$$
	P(x \in C_0)
$$
with the conditional probability
$$
	P(x \in C_0 \mid \#C_0 < \infty).
$$


\begin{center}
\psfragscanon
\includegraphics[scale=0.5]{figures/corrlen_2D_FBC.eps}
\includegraphics[scale=0.5]{figures/corrlen_2D_PBC.eps}

\includegraphics[scale=0.5]{figures/corrlen_3D_FBC.eps}
\includegraphics[scale=0.5]{figures/corrlen_3D_PBC.eps}
\end{center}

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\end{document}
