\begin{figure}[tp]
\centering
\begin{tikzpicture}[x=1cm, y=1cm, font=\scriptsize,
  solidc/.style={line width=0.9pt},
  dashc/.style={line width=0.9pt, dash pattern=on 3pt off 1.5pt}]
\foreach \r/\ph/\st/\th/\ly/\lab in {2.6/0/solidc/38/1.75/{$\mathcal C_0$, height $a_0$},
                             1.95/40/solidc/22/0.95/{$\mathcal C_1$, height $a_1=a_0+\varepsilon_1m_1$},
                             1.35/95/solidc/5/0.15/{$\mathcal C_2$, height $a_2=a_1+\varepsilon_2m_2$},
                             0.8/160/solidc/-30/-0.65/{$\mathcal C_3$, height $a_3=a_2+\varepsilon_3m_3$}}{
  \draw[\st] plot[smooth cycle, samples=36, domain=0:350] ({\r*(1+0.11*sin(3*\x+\ph))*cos(\x)},{0.78*\r*(1+0.11*sin(3*\x+\ph))*sin(\x)});
  \pgfmathsetmacro{\px}{\r*(1+0.11*sin(3*\th+\ph))*cos(\th)}
  \pgfmathsetmacro{\py}{0.78*\r*(1+0.11*sin(3*\th+\ph))*sin(\th)}
  \draw[black!55, line width=0.4pt] (\px,\py) -- (3.45,\ly);
  \node[anchor=west] at (3.5,\ly) {\lab};
}
\fill (0,0) circle (1.4pt) node[below=1pt] {$o$};
\end{tikzpicture}
\caption{The circuit argument for roughness. Around a face $o$ the plane measure has infinitely many
nested open circuits, on each of which the height is constant; $\mathcal C_{i}$ is the outermost one
inside $\mathcal C_{i-1}$ whose height differs from that of $\mathcal C_{i-1}$. By the Markov property and
a reflection symmetry inside each circuit (Lemma~\ref{lem:Markov}), the signs $\varepsilon_i$ of the
height increments between consecutive circuits are independent fair coin flips given everything else.
If $\Pp_c$ were the gradient of a Gibbs measure $\nu$ on heights, the height at $o$ under $\nu$ would be the
height of an outer circuit plus a sum of more and more independent symmetric increments, which cannot
have a tight law (Proposition~\ref{prop:rough}). The same random
walk of circuit heights appears in \cite[Section~4]{GlazmanLammers}.}
\label{fig:circuits}
\end{figure}