From b07d73fdcc4d1e4c38b473ac10779480a4d678ac Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 3 Feb 2022 10:41:10 +0100
Subject: Added new figure highlighting our example function.

---
 figs/action.pdf           | Bin 0 -> 10688 bytes
 figs/stationaryPoints.pdf | Bin 0 -> 8596 bytes
 stokes.tex                |  18 ++++++++++++++++++
 3 files changed, 18 insertions(+)
 create mode 100644 figs/action.pdf
 create mode 100644 figs/stationaryPoints.pdf

diff --git a/figs/action.pdf b/figs/action.pdf
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diff --git a/figs/stationaryPoints.pdf b/figs/stationaryPoints.pdf
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diff --git a/stokes.tex b/stokes.tex
index 37dacbf..8b89104 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -147,6 +147,24 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of
 \emph{all} stationary points of the action, which naturally contains the set of
 \emph{real} stationary points $\Sigma_0$.
 
+\begin{figure}
+  \includegraphics{figs/action.pdf}\hfill
+  \includegraphics{figs/stationaryPoints.pdf}
+
+  \caption{
+    An example of a simple action and its critical points. \textbf{Left:} An
+    action $\mathcal S$ for the $N=2$ spherical (or circular) $3$-spin model,
+    defined for $s_1,s_2\in\mathbb R$ on the circle $s_1^2+s_2^2=2$ by
+    $\mathcal S(s_1,s_2)=1.051s_1^3+1.180s_1^2s_2+0.823s_1s_2^2+1.045s_2^3$. In
+    the example figures in this section, we will mostly use the single angular
+    variable $\theta=\arctan(s_1,s_2)$, which parameterizes the unit circle and
+    its complex extension. \textbf{Right:} The stationary points of $\mathcal
+    S$ in the complex-$\theta$ plane. In this example,
+    $\Sigma=\{\blacklozenge,\bigstar,\blacktriangle,\blacktriangledown,\bullet,\blacksquare\}$
+    and $\Sigma_0=\{\blacklozenge,\blacktriangledown\}$.
+  }
+\end{figure}
+
 Assuming $\mathcal S$ is holomorphic (and that the phase space $\Omega$ is
 orientable, which is usually true) the integral in \eref{eq:partition.function}
 can be considered an integral over a contour in the complex phase space $\tilde\Omega$,
-- 
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