From 7ac614ae424854e032decd28b1de6cd71ea4d109 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 25 Mar 2022 15:53:05 +0100
Subject: Added reference to new review paper.

---
 stokes.tex | 20 ++++++++++++++++++--
 1 file changed, 18 insertions(+), 2 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index b2176ce..ba5a11e 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -101,7 +101,7 @@ divergent partition function, and can only be properly defined by continuation
 from a parameter regime where everything is well-defined \cite{}. Others result
 in oscillatory phase space measures that spoil the use of Monte Carlo or saddle
 point techniques, but can be treated in a regime where the measure does not
-oscillated and the results continued to the desired model \cite{}.
+oscillated and the results continued to the desired model \cite{Alexandru_2022_Complex}.
 
 In any case, the nicest modern technique (which we will describe in some detail
 later) consists of deforming the phase space integral into a complex phase
@@ -157,6 +157,7 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of
 \emph{real} stationary points $\Sigma_0$.
 
 \begin{figure}
+  \hspace{5pc}
   \includegraphics{figs/action.pdf}\hfill
   \includegraphics{figs/stationaryPoints.pdf}
 
@@ -194,9 +195,12 @@ without changing their value. This means that we are free to choose a nicer
 contour than our initial phase space $\Omega$.
 
 \begin{figure}
+  \hspace{5pc}
   \includegraphics{figs/hyperbola_1.pdf}\hfill
   \includegraphics{figs/hyperbola_2.pdf}\hfill
-  \includegraphics{figs/hyperbola_3.pdf}\\
+  \includegraphics{figs/hyperbola_3.pdf}
+
+  \hspace{5pc}
   \includegraphics{figs/anglepath_1.pdf}\hfill
   \includegraphics{figs/anglepath_2.pdf}\hfill
   \includegraphics{figs/anglepath_3.pdf}
@@ -264,6 +268,7 @@ of the thimbles must represent the same element of this relative homology
 group.
 
 \begin{figure}
+  \hspace{5pc}
   \includegraphics{figs/thimble_homology.pdf}
   \hfill
   \includegraphics{figs/antithimble_homology.pdf}
@@ -316,6 +321,7 @@ two stationary points are called \emph{Stokes lines}. An example of this
 behavior can be seen in Fig.~\ref{fig:1d.stokes}.
 
 \begin{figure}
+  \hspace{5pc}
   \includegraphics{figs/thimble_stokes_1.pdf}\hfill
   \includegraphics{figs/thimble_stokes_2.pdf}\hfill
   \includegraphics{figs/thimble_stokes_3.pdf}
@@ -339,6 +345,7 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}.
 \end{figure}
 
 \begin{figure}
+  \hspace{5pc}
   \includegraphics{figs/thimble_orientation_1.pdf}\hfill
   \includegraphics{figs/thimble_orientation_2.pdf}\hfill
   \includegraphics{figs/thimble_orientation_3.pdf}
@@ -431,6 +438,8 @@ to $z$, i.e., $z^\dagger u=0$, $Pu=u$, the identity.
 
 
 \begin{figure}
+  \hspace{5pc}
+  \hfill
   \includegraphics{figs/thimble_flow.pdf}
 
   \caption{Example of gradient descent flow on the action $\mathcal S$ featured
@@ -677,6 +686,8 @@ $\blacktriangle$ have exactly the same imaginary energy, and yet they do not
 share a thimble.
 
 \begin{figure}
+  \hspace{5pc}
+  \hfill
   \includegraphics{figs/6_spin.pdf}
   \caption{
   Some thimbles of the circular 6-spin model, where the argument of $\beta$ has
@@ -693,6 +704,8 @@ thimbles is the same as the codimension of the constant imaginary energy
 surface is one, and such a surface can divide space into regions. However, in higher dimensions thimbles do not have codimension high enough to divide space into regions.
 
 \begin{figure}
+  \hspace{5pc}
+  \hfill
   \includegraphics{figs/2_spin_thimbles.pdf}
   \caption{
     Thimbles of the $N=3$ spherical 2-spin model projected into the
@@ -1474,7 +1487,9 @@ along with the inequality produce the required bound on $|\delta|$ as a
 function of $\Delta$ and $\arg\delta$.
 
 \begin{figure}
+  \hspace{5pc}
   \includegraphics{figs/bound.pdf}
+  \hfill
   \includegraphics{figs/example_bound.pdf}
 
   \caption{
@@ -1565,6 +1580,7 @@ in $\beta$, which would be equivalent to allowing non-constant terms in the
 Jacobian over the thimble.
 
 \begin{figure}
+  \hspace{5pc}
   \hfill\includegraphics{figs/obuchi_3-spin.pdf}
   \caption{
     Phases of the 3-spin model in the complex-$\beta$, following Obuchi \&
-- 
cgit v1.2.3-70-g09d2

