
Here we elaborate on the Fourier methods of \cite{bib:bu,bib:qmath,bib:bu08}:
in particular section 5 of each of those three papers.  These are used to
establish some theoretical results in sections xxx, xxx, and xxx.

%xxx this is at present very much a work in progress --- a scratch space of
%formulas, propositions, etc.

%% ================================================================
\section{Lattice and dual space}

Our goal for this section is to describe a probability model on the dual state
space, and write expectations of class-function random variables in terms of
that probability model.  These will enable us to obtain expressions for those
expectations, for points distributed on the continuum.  For clarity of
exposition, we permit some overlap with definitions from section
\ref{sec:model_defns}.

\begin{defn}
We define the \bfidx{lattice} and \bfidx{dual lattice} to be
$$
	\Lambda = [0,L]^3
	\qqand
	\Lambda^* = \left(\Z/L\right)^3
$$
where the former is taken to have \plainidx{periodic boundary conditions}.
We write
$$
	\vecX = (\vecx_1, \ldots, \vecx_N) \in \Lambda^N
	\qqand
	\vecK = (\veck_1, \ldots, \veck_N) \in (\Lambda^*)^N.
$$
The \bfidx{state space} and \bfidx{dual state space}
of the model of spatial permutations are
$$
	\Omega_{\Lambda,N} = \Lambda^N \times \SN
	\qqand
	\Omega^*_{\Lambda^*,N} = (\Lambda^*)^N \times \SN
$$
where $\SN$ is the group of \plainidx{permutations} of $N$ points.
\end{defn}

\begin{defn}
A \bfidx{Fourier mode} is any $\veck\in\Lambda^*$;
a \bfidx{set of Fourier modes} is any $\vecK\in(\Lambda^*)^N$.
\end{defn}

%% ================================================================
\section{Occupation numbers}

\begin{defn}
An \bfidx{occupation number} is a three-dimensional sequence of $\vecn$
integers indexed by $\veck\in\Lambda^*$.  That is, $\vecn$ simply assigns an
integer count to each $\veck$.  An element of this sequence, i.e.  the integer
corresponding to a fixed $\veck$, is written $n_\veck$.  The \bfidx{set of all
possible occupation numbers} is written $\mcN_{\Lambda}$.  We define the set of
occupation numbers with total number $N$:
$$
	\mcN_{\Lambda,N}(\vecn) = \left\{
		\vecn \in \mcN_{\Lambda} : \sum_{\veck\in\Lambda^*} n_\veck = N
	\right\}.
$$
\end{defn}

\begin{defn}
The map $\phi: (\Lambda^*)^N \to \mcN$, written
$$
	\vecn(\vecK) = \phi(\vecK),
$$
counts the number of times a given $\veck$ occurs in $\vecK$.  That is,
$\phi(\vecK) = \vecn(\veck_1, \ldots, \veck_N)$ where
\begin{align*}
	n_\veck &= \#\{i=1,\ldots,N : \veck_i = \veck\}
\end{align*}
\end{defn}

\beginex
If $N=4$ and
$$
	\vecK = (\veck_1, \veck_2, \veck_3, \veck_4)
	= ((0,0,0), (0,0,1), (0,-2,0),(0,0,0)),
$$
then $\vecn(\vecK)$ has
\begin{align*}
	n_{(0,0,0)}  &= 2, &
	n_{(0,0,1)}  &= 1, &
	n_{(0,-2,0)} &= 1, \qand&
	n_\veck      &= 0
\end{align*}
for all other $\veck$'s.  Note that
$$
	\vecK = (\veck_1, \veck_2, \veck_3, \veck_4)
	= ((0,0,0), (0,0,0), (0,0,1),(0,-2,0))
$$
gives the same $\vecn(\vecK)$.
\mendex

\begin{defn}
The set of all $\vecK$'s mapping to a given $\vecn$ is written
$$
	\mcK(\vecn) = \phi^{-1}(\vecn).
$$
A set of Fourier modes $\vecK$ is said to be \bfidx{compatible with} $\vecn$
if $\vecK \in \mcK(\vecn)$, i.e. if $\phi(\vecK) = \vecn$.
\end{defn}

\begin{prop}
The map $\phi: \vecK\mapsto\vecn(\vecK)$ is onto but not one-to-one.  The
number of $\vecK$'s that are sent to a given $\vecn$ is equal to
\begin{align*}
	\#\mcK(\vecn) &= \frac{N!}{\prod_{\veck\in\Lambda^*} n_{\veck}!}.
\end{align*}
\begin{proof}
{}[xxx elaborate on this.] This is simply a multinomial coefficient: all
possible permutations of $\veck_1,\ldots,\veck_N$ divided by the number of ways
of having duplicates.
\end{proof}
\end{prop}

%% ================================================================
\section{Fourier transforms}

\begin{defn}
\label{defn:xi_veps}
Define $\xi,\veps:\R^d \to \R$ and $C \in \R$ by
\begin{align*}
	\xi(\vecx) &= \|\vecx\|^2, &
	C &= (4\pi\beta)^{3/2}, \qand &
	\veps(\veck) &= 4 \pi^2 \beta \|\veck\|^2.
\end{align*}
\end{defn}

\begin{defn}
\label{defn:periodization}
For real-valued $f: L^1(\R^3)$ we define the \bfidx{periodization} of $f$ to be
\begin{align*}
	f_\Lambda(\vecx) &= \sum_{\vecy \in \Z^3} f(\vecx + L \vecy).
\end{align*}
Notice that $f_\Lambda \in L^1(\Lambda)$.
\end{defn}

\begin{defn}
\label{defn:xi_periodization}
For $\xi$ we make an exception to definition \ref{defn:periodization}:
we periodize $\xi$ by defining $\xi_\Lambda: \Lambda \to \R$ to be
\begin{align*}
	e^{-\xi_\Lambda(\vecx)} &= \sum_{\vecy \in \Z^3} e^{-\xi(\vecx + L \vecy)}.
\end{align*}
\end{defn}

\begin{defn}
xxx \bfidx{Fourier transform}.
For $f(\vecx)$ on $\Lambda$,
$$
	\fhat(\veck) = \int_\Lambda e^{-i 2 \pi \veck\cdot \vecx} f(\vecx)\,d\vecx
$$
with $\vecx \in\Lambda$ and $\veck \in \Lambda^*$.
xxx Integrability conditions on $f$.
\end{defn}

\begin{defn}
As with definitions \ref{defn:periodization} and \ref{defn:xi_periodization},
we have a modified definition of the Fourier transform of $\xi$.
The constant $C$ and function $\veps$ of definition \ref{defn:xi_veps} were
chosen to satisfy
\begin{align*}
	C e^{-\veps(\veck)}
	&= \int_{\R^3} e^{-i 2 \pi \veck\cdot \vecx} e^{\xi(-\vecx)}\,d\vecx.
\end{align*}
Note that $\int_{\R^d} e^{-\veps(\veck)}\,d\veck = 1/C$.
\end{defn}

%% ================================================================
\section{Probability models} 

Note that the true probability model on the dual state space is not the na\"ive
one stated here.  The correct one is in the next section.

\begin{defn}
\begin{align}
\label{eqn:Fourier_RCM_H}
	H(\vecX,\pi) &= 
	\sum_{i=1}^N \xi(\vecx_i-\vecxpii) +
	\sum_{\ell=1}^N \alpha_\ell r_\ell(\pi)
	%
	&\qand
	%
	H^*(\vecK,\pi) &= 
	\sum_{i=1}^N \veps(\veck_i) +
	\sum_{\ell=1}^N \alpha_\ell r_\ell(\pi). \\
%
	H_\Lambda(\vecX,\pi) &=
	\sum_{i=1}^N \xi_\Lambda(\vecx_i-\vecxpii) +
	\sum_{\ell=1}^N \alpha_\ell r_\ell(\pi)
	%
	&
	%
	&
\end{align}
where $r_\ell(\pi)$ counts the number of $\ell$-cycles in $\pi$ and
$\alpha_\ell$ is a \bfidx{cycle weight}.
\end{defn}

\index{partition function}Partition functions, for a fixed point configuration
$\vecX$ and for an average over point configurations, respectively:
\begin{align*}
	Y(\Lambda,\vecX) &= \sum_{\sigma\in\SN} e^{-H(\vecX,\sigma)}
	&\qand
	Y^*(\Lambda,\vecK) &= \sum_{\sigma\in\SN} e^{-H^*(\vecK,\sigma)}
	\\
%
	Z(\Lambda,N) &= \frac{1}{N!} \int_{\Lambda^N}
		Y(\Lambda,\vecX)\,d\vecX.
	&\qand
	Z^*(\Lambda,N) &= \frac{1}{N!} \int_{(\Lambda^*)^N}
		Y^*(\Lambda,\vecK)\,d\vecK.
\end{align*}

\begin{defn}
xxx definition?  Or proposition??
\begin{align*}
	Z^*(\Lambda,N) &= \frac{1}{C^N} Z(\Lambda,N).
\end{align*}
\end{defn}

Then we have two corresponding discrete probability measures\index{probability
measure} on the finite set $\SN$, for a fixed point configuration $\vecX$ and
for an average over point configurations, respectively:
\begin{align*}
	P_{\Lambda}(\vecX,\pi)
	&= \frac{e^{-H(\vecX,\pi)}}{\sum_{\sigma\in\SN} e^{-H(\vecX,\sigma)}}
	&
	P^*_{\Lambda}(\vecK,\pi)
	&= \frac{e^{-H^*(\vecK,\pi)}}{\sum_{\sigma\in\SN} e^{-H^*(\vecK,\sigma)}}
	\\
%
	P_{\Lambda,N}(\pi)
	&= \frac
		{\frac{1}{N!} \int_{\Lambda^N}d\vecX e^{-H(\vecX,\pi)}}
		{\frac{1}{N!} \int_{\Lambda^N}d\vecX \sum_{\sigma\in\SN}
			e^{-H(\vecX,\sigma)}}
	&
	P^*_{\Lambda,N}(\pi)
	&= \frac
		{\frac{1}{N!} \int_{(\Lambda*)^N} d\vecK e^{-H^*(\vecK,\pi)}}
		{\frac{1}{N!} \int_{(\Lambda*)^N} d\vecK \sum_{\sigma\in\SN}
			e^{-H^*(\vecK,\sigma)}}
\end{align*}

These may also be written in terms of the partition functions as
\begin{align}
\label{eqn:fourappx_RCM_P}
	P_{\Lambda}(\vecX,\pi)
	&= \frac{e^{-H(\vecX,\pi)}}{Y(\Lambda,\vecX)}
	&
	P^*_{\Lambda}(\vecK,\pi)
	&= \frac{e^{-H^*(\vecK,\pi)}}{Y^*(\Lambda^*,\vecK)}
	\\
%
	P_{\Lambda,N}(\pi)
	&= \frac{\int_{\Lambda^N}d\vecX e^{-H(\vecX,\pi)}}{Z(\Lambda,N)N!}.
	&
	P^*_{\Lambda,N}(\pi)
	&= \frac{\int_{(\Lambda^*)^N}d\vecK e^{-H^*(\vecK,\pi)}}{Z^*(\Lambda,N)N!}.
\end{align}

For\index{expectation} a random variable $\theta(\pi)$, we have
$$
	\bbE_{\Lambda,\vecX}[\theta] = \frac{1}{Y(\vecX,\Lambda)}
		\sum_{\pi\in\SN} \theta(\pi) e^{-H(\vecX,\pi)}
	\qand
	\bbE_{\Lambda,N}[\theta] = \frac{1}{Z(\Lambda,N) N!}
	\int_{\Lambda^N} d\vecX \sum_{\pi\in\SN} \theta(\pi) e^{-H(\vecX,\pi)}.
$$
In either case, we also write the expectation as
\begin{align}
\label{eqn:fourappx_E_pi}
	\bbE_\pi[\theta(\pi)] = \sum_{\pi\in\SN} P(\pi) \theta(\pi).
\end{align}

%% ================================================================
\section{Actual probability model for the dual state space} 

xxx define the probability model on the dual space
so that \bfidx{marginal distributions} coincide:
\begin{align*}
	P_{\Lambda}(\Lambda^N, \{\pi\})
	&=
	P^*_{\Lambda}((\Lambda^*)^N, \{\pi\}).
\end{align*}
xxx i.e. if $\vecK$ is constant on cycles of $\pi$.  Motivate this by referring
back to symmetrized basis functions.

xxx elaborate:
\begin{align*}
	P^*_{\Lambda}(\vecK, \pi) &= \begin{cases}
		\frac{1}{Z^*(\Lambda,N) N!} \, e^{-H^*(\vecK,\pi)} &
		\textrm{if $\veck_i = \veck_{\pi(i)}$ for all $i$}, \\
		%
		0 &
		\textrm{otherwise}.
	\end{cases}
\end{align*}
xxx write out: $Z^*$ is the sum over all such numerators.

xxx goals:
\begin{itemize}

\item Sum over permutations and then over Fourier modes compatible with
occupation numbers to obtain an expression for $Z^*$.  From there, obtain the
nice expression for $\bbE$ for class-function $\theta$.

\end{itemize}

xxx summing over permutations,
\begin{align*}
	P^*_{\Lambda}(\vecK, \SN) &=
		\frac{1}{Z^*(\Lambda,N) N!} \,
		e^{-\sum_{\veck\in\Lambda^*}\veps(\veck)}h_{n_\veck}
		\prod_{\veck\in\Lambda^*} n_\veck!
\end{align*}
with [xxx] $n_\veck$ being elements of the sequence $\vecn(\vecK)$.

xxx have defined
$$
	h_n = \frac{1}{n!} e^{-\sum_{\ell=1}^n \alpha_\ell r_\ell(\pi)}.
$$

xxx summing over Fourier modes which are compatible with occupation numbers
yields
\begin{align*}
	P^*_{\Lambda}(\mcK(\vecn), \SN) &=
		\frac{1}{Z^*(\Lambda,N) N! }\,
		e^{-\sum_{\veck\in\Lambda^*}\veps(\veck)n_\veck}
\end{align*}

xxx ``it follows that''
\begin{align*}
	Z^*(\Lambda,N) &= \sum_{\vecn \in \mcN_{\Lambda,\vecn}}
		e^{-\sum_{\veck\in\Lambda^*} \veps(\veck) n_\veck}.
\end{align*}

xxx BU lemma 5.4: for class-function $\theta(\pi)$,
\begin{align*}
	\bbE_{\Lambda,N}[\theta] &=
	\sum_{\vecn\in\mcN_{\Lambda,N}} P_{\Lambda,N}(\vecn)
	\sum_{\pi\in\SN} \theta(\pi) P_{\Lambda,N}(\pi \mid \veck)
\end{align*}
where [xxx] $\vecK$ is any element of $\mcK(\vecn)$.

xxx by the corollary (BU ppr)
\begin{align*}
	\bbE_{\Lambda,N}[\theta]
	&= \frac{1}{Z^*(\Lambda,N) N!}
	\sum_{\pi\in\SN} \theta(\pi)
	\sum_{\veck_1,\ldots,\veck_N \in \Lambda^*}
	\exp\left\{-\sum_{i=1}^N \veps(\veck_i)\right\}
	\prod_{i=1}^N \delta_{\veck_i,\veck_{\pi(i)}}.
\end{align*}

%% ================================================================
\section{Results}

% xxx separate section for proofs?

xxx attribute to relevant lemmas/corollaries/etc. of BU and BU08.  I am filling
in the details.  Merge BU and BU08 results.

\begin{lem}
For $f \in L^1(\R^3)$ and all $n=1,2,\ldots$,
\begin{align*}
	\int_{\Lambda^n} d\vecx_1 \cdots d\vecx_n
	\prod_{i=1}^n f_\Lambda(\vecx_i - \vecx_{i-1})
	&= \sum_{\veck \in \Lambda^*} \fhat(\veck)^n
\end{align*}
where $\vecx_0 := \vecx_n$.
\begin{proof}
xxx type me up.
\end{proof}
\end{lem}

\begin{cor}
For all $\pi\in\SN$,
\begin{align*}
	\int_{\Lambda^N} d\vecx_1 \cdots d\vecx_N
		\prod_{i=1}^N f_\Lambda(\vecx_i - \vecx_{\pi(i)}
	&= \sum_{\veck_1, \ldots, \veck_N \in \Lambda^*}
		\prod_{i=1}^N \delta_{\veck_i,\veck_{\pi(i)}}
		\prod_{j=1}^N \fhat(\veck_j).
\end{align*}
\begin{proof}
xxxx type me up.
\end{proof}
\end{cor}

\begin{prop}
For a random variable $\theta(\pi)$ on $\Omega_{\Lambda,N}$,
\begin{align*}
	\bbE_{\Lambda,N}[\theta]
	&=
	\frac{1}{Z'(\Lambda,N) N!}
	\sum_{\pi\in\SN} \theta(\pi)
	\sum_{\veck_1, \ldots, \veck_N \in \Lambda^*}
	\exp\left\{\sum_{i=1}^N \veps(\veck_i)\right\}
	\prod_{i=1}^N \delta_{\veck_i,\veck_{\pi(i)}}.
\end{align*}
\begin{proof}
xxx type me up, using the corollary.
\end{proof}
\end{prop}

xxx def class functions, here or higher.

\begin{prop}
For all class-function random variables $\theta(\pi)$,
\begin{align*}
	\bbE_{\Lambda,N}(\theta)
	&= \sum_{\vecn \in \mcN_{\Lambda,N}} P_{\Lambda,N}(\vecn)
	\sum_{\pi\in\SN} \theta(\pi) P_{\Lambda,N}(\theta\mid\vecK),
\end{align*}
where $\vecK$ is any element of $\mcK(\vecn)$.
\begin{proof}
xxx type me up.
\end{proof}
\end{prop}
