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.
---
figs/ground_complexity.pdf | Bin 0 -> 126731 bytes
figs/im_complexity.pdf | Bin 0 -> 209597 bytes
figs/leg_complexity.pdf | Bin 0 -> 6424 bytes
figs/re_complexity.pdf | Bin 0 -> 109687 bytes
stokes.bib | 14 ++++++++++++++
stokes.tex | 37 ++++++++++++++++++++++++++++++++++++-
6 files changed, 50 insertions(+), 1 deletion(-)
create mode 100644 figs/ground_complexity.pdf
create mode 100644 figs/im_complexity.pdf
create mode 100644 figs/leg_complexity.pdf
create mode 100644 figs/re_complexity.pdf
diff --git a/figs/ground_complexity.pdf b/figs/ground_complexity.pdf
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diff --git a/figs/im_complexity.pdf b/figs/im_complexity.pdf
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diff --git a/figs/leg_complexity.pdf b/figs/leg_complexity.pdf
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diff --git a/figs/re_complexity.pdf b/figs/re_complexity.pdf
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diff --git a/stokes.bib b/stokes.bib
index 70067ee..6ee17a2 100644
--- a/stokes.bib
+++ b/stokes.bib
@@ -299,6 +299,20 @@
doi = {10.1103/physrevlett.92.240601}
}
+@article{Howls_1997_Hyperasymptotics,
+ author = {Howls, C. J.},
+ title = {Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem},
+ journal = {Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences},
+ publisher = {The Royal Society},
+ year = {1997},
+ month = {11},
+ number = {1966},
+ volume = {453},
+ pages = {2271--2294},
+ url = {https://doi.org/10.1098%2Frspa.1997.0122},
+ doi = {10.1098/rspa.1997.0122}
+}
+
@article{Kac_1943_On,
author = {Kac, M.},
title = {On the average number of real roots of a random algebraic equation},
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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