\begin{figure}[tp]
\centering
\begin{tikzpicture}[x=0.62cm, y=0.5cm, font=\scriptsize, >={Stealth[length=1.5mm]}]
% left panel
\fill[black!7] (0,2) rectangle (6.4,4);
\foreach \y/\lab in {0/{0},2/{M},4/{2M}}{
  \draw[dashed, black!50] (0,\y) -- (6.4,\y);
  \node[anchor=east] at (0,\y) {$\lab$};}
\node[anchor=west, black!60] at (6.45,1) {$k$ even};
\node[anchor=west, black!60] at (6.45,3) {$k$ odd};
\node[anchor=west, black!60] at (6.45,5) {$k$ even};
\draw[line width=0.9pt] plot coordinates {(0,0.5) (0.5,1.2) (1,0.9) (1.5,1.7) (1.714,2) (2,2.4) (2.5,2.1) (3,2.9) (3.5,3.6) (4,3.3) (4.389,4) (4.5,4.2) (5,4.8) (5.5,4.4) (6,5.2)};
\draw[->] (0,-0.4) -- (6.6,-0.4) node[below left] {path in $\Z^2$};
\node[anchor=west] at (0.1,5.6) {$D$};
% arrow
\draw[->, line width=0.8pt] (8.2,1.1) -- (9.8,1.1) node[midway, above] {fold $R$};
% right panel
\begin{scope}[shift={(10.6,0)}]
\foreach \y/\lab in {0/{0},2/{M}}{
  \draw[dashed, black!50] (0,\y) -- (6.4,\y);
  \node[anchor=east] at (0,\y) {$\lab$};}
\draw[line width=0.9pt] plot coordinates {(0,0.5) (0.5,1.2) (1,0.9) (1.5,1.7) (1.714,2) (2,1.6) (2.5,1.9) (3,1.1) (3.5,0.4) (4,0.7) (4.389,0) (4.5,0.2) (5,0.8) (5.5,0.4) (6,1.2)};
\draw[->] (0,-0.4) -- (6.6,-0.4) node[below left] {path in $\Z^2$};
\node[anchor=west] at (0.1,2.6) {$D'$};
\end{scope}
\end{tikzpicture}
\caption{The fold, schematically. On the left, the difference $D$ along a path, with the bands of
levels $[kM,(k+1)M)$ shaded alternately. The fold shifts the two copies on every band and also exchanges
them on the odd bands, so that $D'=D-kM$ on even bands and $D'=(k+1)M-D$ on odd bands: the difference is folded like an
accordion into a strip of width about $M$ (right). In the proof the band index $k(x)$ of a face is
defined through the infinite minus clusters at the levels $jM$, not through the value of $D$ at $x$
alone, which is what makes the fold admissible and keeps the joint weight unchanged
(Lemmas~\ref{lem:band} and~\ref{lem:fold}).}
\label{fig:fold}
\end{figure}