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\section{Various energy functions}

Here we tabulate, for handy reference, various energy functions used
in \cite{bib:gru} and \cite{bib:qmath}.

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\begin{tabular}{|l|l|}
\hline &\\
	Permutation energy &
	$ H_\Lambda(\pi) = \sum_{x \in \Lambda} |x-\pi(x)|^2 $ \\
	& \\

\hline &\\
	Permutation/point probability contribution &
	$ Q(\pi,x) = e^{-\alpha|x-\pi(x)|^2} $ \\
	& \\

\hline &\\
	Permutation probability numerator &
	$ P^*_\Lambda(\pi) = \prod_{x\in\Lambda} Q(\pi,x) $ \\

	&
	$ = \prod_{x\in\Lambda} e^{-\alpha|x-\pi(x)|^2} $ \\

	&
	$	= e^{-\alpha \sum_{x \in\Lambda} |x-\pi(x)|^2} $ \\

	&
	$	= e^{-\alpha H_\Lambda(\pi)} $ \\
	& \\

\hline &\\
	Partition function &
	$ Z_\Lambda = \sum_{\pi\in\bbB_\Lambda} P^*_\Lambda(\pi) $ \\
	& \\

\hline &\\
	Permutation probability &
	$
		P_\Lambda(\pi) = \frac{P^*_\Lambda(\pi)}{Z_\Lambda}
		= \frac{e^{-\alpha H_\Lambda(\pi)}}{Z_\Lambda}
	$ \\
	& \\

\hline &\\
	Probability the origin is in a cycle of length $k$ &
	$
		P_\Lambda(\ell_0=k) = \sum_{\pi\in\bbB_\Lambda: \ell_0=k} P_\Lambda(\pi)
	$ \\

	&
	$P(\ell_0=k) = \lim_{\Lambda\upto\Z^d} P_\Lambda(\ell_0=k)$ \\
	& \\

\hline &\\
	Probability the origin is in an infinite cycle &
	$ \phi(\alpha) = 1 - \sum_{k=1}^\infty P(\ell_0=k)
	$ \\
	& \\

\hline &\\
	Thermodynamic potential &
	$ f_\Lambda(\alpha) = \frac{\log(Z_\Lambda)}{\mid \Lambda \mid}
	$ \\

	&
	$ f(\alpha) = \lim_{\Lambda\upto\Z^d} f_\Lambda(\alpha)
	$ \\
	& \\

\hline
\end{tabular}
