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\section{$\Delta V$ for two-jump interactions}

Recall from \cite{bib:qmath} that we define the two-body interaction potential
for a permutation $\pi$ to be
\begin{align*}
	V_\pi &= \sum_{u<v} V(u, \pi(u), v, \pi(v))
\end{align*}
where $u$, $v$ are lattice sites.

For an implementation of a Metropolis algorithm, one wishes to compute the
change in potential energy when one sends $\pi$ to $\pi'$ via
$$
	\pi:
	\begin{pmatrix}
		x & \iy & u & \cdots\\
		\downarrow & \downarrow & \downarrow\\
		\ix & y & \iu & \cdots
	\end{pmatrix}
	\qquad
	\pi':
	\begin{pmatrix}
		x & \iy & u & \cdots\\
		\downarrow & \downarrow & \downarrow\\
		y & \ix & \iu & \cdots
	\end{pmatrix}
$$
where $\pi'(u) = \pi(u)$ for all $u \ne x, \iy$.  One has
\begin{align*}
	V_{\pi'} - V_\pi
	&=	\sum_{u<v} V(u, \pi'(u), v, \pi'(v))
	-	\sum_{u<v} V(u, \pi(u), v, \pi(v)).
\end{align*}
Since $\pi'(u) = \pi(u)$ for all $u \ne x, \iy$, we may rewrite this in terms
of $\pi$ only as
\begin{align*}
&= \sum_{{u<x,}  \atop{u\ne \iy}}\bigg(V(u,  \iu,x,  y)  - V(u,  \iu,x, \ix) \bigg) \\
&+ \sum_{{u>x,}  \atop{u\ne \iy}}\bigg(V(x,  y,  u,  \iu)- V(x,  \ix,u, \iu) \bigg) \\
&+ \sum_{{u<\iy,}\atop{u\ne x}}  \bigg(V(u,  \iu,\iy,\ix)- V(u,  \iu,\iy,y) \bigg) \\
&+ \sum_{{u>\iy,}\atop{u\ne x}}  \bigg(V(\iy,\ix,u,  \iu)- V(\iy, y,  u, \iu) \bigg) \\
%
&+ \chi_{x<\iy} \bigg(V(x,   y, \iy, \ix) -  V(x,  \ix, \iy,  y) \bigg) \\
&+ \chi_{\iy<x} \bigg(V(\iy, \ix, x,   y) -  V(\iy,  y,  x, \ix) \bigg).
\end{align*}
This expression is clumsy.  Recall, however, that the interaction potential satisfies
the symmetry condition
$$
	V(x,x',y,y') = V(y,y',x,x').
$$
Then we have simply
\boxalign{
	V_{\pi'} - V_\pi
&= \sum_{{u\ne x,} \atop{u\ne \iy}} \bigg(
		V(u,   \iu, x,   y)
	-	V(u,   \iu, x,   \ix) \bigg) \\
&+ \sum_{{u\ne x,} \atop{u\ne \iy}} \bigg(
		V(u,   \iu, \iy, \ix)
	-	V(u,   \iu, \iy, y) \bigg) \\
%
	&+	V(x,   y,    \iy,  \ix)
	-	V(x,  \ix, \iy,  y).
}
