From 506ca083dcf5f7df88cc5d489b11d32d0f2e43c1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Feb 2022 14:25:24 +0100
Subject: Replaced schematic local flow with figure of global flow for the
 example action.

---
 figs/thimble_flow.pdf | Bin 0 -> 144566 bytes
 stokes.tex            |  22 +++++++++++++---------
 2 files changed, 13 insertions(+), 9 deletions(-)
 create mode 100644 figs/thimble_flow.pdf

diff --git a/figs/thimble_flow.pdf b/figs/thimble_flow.pdf
new file mode 100644
index 0000000..5499a5b
Binary files /dev/null and b/figs/thimble_flow.pdf differ
diff --git a/stokes.tex b/stokes.tex
index 3c8af84..aa11dc3 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -549,6 +549,19 @@ manifold: its tangent at any point $z$ is given by $\partial(z^Tz)=z$, and
 $Pz=z-z|z|^2/|z|^2=0$. For any vector $u$ perpendicular to $z$, i.e.,
 $z^\dagger u=0$, $Pu=u$.
 
+\begin{figure}
+  \includegraphics{figs/thimble_flow.pdf}
+
+  \caption{Example of gradient descent flow on the action $\mathcal S$ featured
+    in Fig.~\ref{fig:example.action} in the complex-$\theta$ plane, with
+    $\arg\beta=0.4$. Symbols denote the stationary points, while thick blue and
+    red lines depict the thimbles and antithimbles, respectively. Streamlines
+    of the flow equations are plotted in a color set by their value of
+    $\operatorname{Im}\beta\mathcal S$; notice that the color is constant along
+    each streamline.
+  } \label{fig:flow.example}
+\end{figure}
+
 Gradient descent on $\operatorname{Re}\beta\mathcal S$ is equivalent to
 Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\beta\mathcal S$
 and conjugate coordinates given by the real and imaginary parts of each complex
@@ -616,15 +629,6 @@ $C\neq\operatorname{Im}\mathcal S(z_0)$ is split into two disconnected pieces, o
 lying in each of two quadrants corresponding with its value relative to that at
 the critical point.
 
-\begin{figure}
-  \includegraphics{figs/local_flow.pdf}
-  \caption{
-    Gradient descent in the vicinity of a critical point, in the $z$--$z^*$
-    plane for an eigenvector $z$ of $(\partial\partial\mathcal S)^\dagger P$. The flow
-    lines are colored by the value of $\operatorname{Im}H$.
-  } \label{fig:local_flow}
-\end{figure}
-
 Continuing to `insert' critical points whose imaginary energy differs from $C$,
 one repeatedly partitions the space this way with each insertion. Therefore,
 for the generic case with $\mathcal N$ critical points, with $C$ differing in
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