From 8ebd1ca5e88cfaf2ff8d7aa40a383c01997fa438 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Mar 2022 17:20:48 +0100
Subject: Added some new figures for the p-spin complexity.

---
 stokes.tex | 37 ++++++++++++++++++++++++++++++++++++-
 1 file changed, 36 insertions(+), 1 deletion(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 0973242..4a5f3d5 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -749,7 +749,9 @@ surface is one, and such a surface can divide space into regions. However, in hi
 
 After all the work of decomposing an integral into a sub over thimbles, one
 eventually wants to actually evaluate it. For large $|\beta|$ and in the
-absence of any Stokes points, one can come to a nice asymptotic expression.
+absence of any Stokes points, one can come to a nice asymptotic expression. For
+thorough account of evaluating these integrals (including \emph{at} Stokes
+points), see Howls \cite{Howls_1997_Hyperasymptotics}.
 
 Suppose that $\sigma\in\Sigma$ is a stationary point at $s_\sigma\in\tilde\Omega$
 with a thimble $\mathcal J_\sigma$ that is not involved in any upstream Stokes points.
@@ -1388,7 +1390,40 @@ $\operatorname{Re}u<0$ then $\operatorname{Re}\sqrt{u^2-1}>0$. If the real part
 is zero, then the sign is taken so that the imaginary part of the root has the
 opposite sign of the imaginary part of $u$.
 
+\cite{Auffinger_2012_Random}
 
+\begin{figure}
+  \hspace{4pc}
+  \includegraphics{figs/re_complexity.pdf}
+  \hspace{-2pc}
+  \includegraphics{figs/im_complexity.pdf}
+  \includegraphics{figs/leg_complexity.pdf}
+
+  \caption{
+    The complexity of the 3-spin spherical model in the complex plane, as a
+    function of pure real and imaginary energy (left and right) and the
+    magnitude $(\operatorname{Im}s)^2/N$ of the distance into the complex
+    configuration space. The thick black contour shows the line of zero
+    complexity, where stationary points become exponentially rare in $N$.
+  } \label{fig:p-spin.complexity}
+\end{figure}
+
+\begin{figure}
+  \hspace{2pc}
+  \includegraphics{figs/ground_complexity.pdf}
+
+  \caption{
+    The complexity of the 3-spin spherical model in the complex plane, as a
+    function of pure real energy and the magnitude $(\operatorname{Im}s)^2/N$
+    of the distance into the complex configuration space. The thick black
+    contour shows the line of zero complexity, where stationary points become
+    exponentially rare in $N$. The shaded region shows where stationary points
+    have a gapped spectrum. The complexity of the 3-spin model on the real
+    sphere is shown below the horizontal axis; notice that it does not
+    correspond with the limiting complexity in the complex configuration space
+    below the threshold energy.
+  } \label{fig:ground.complexity}
+\end{figure}
 
 \subsection{Pure \textit{p}-spin: where are my neighbors?}
 
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