From 6ebfaf80ca120c0b623ff18520fd7aa15efcea3f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 3 Feb 2022 11:14:13 +0100
Subject: Made figure captions more consistent, and added details about the
 phase of β.
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 stokes.tex | 56 ++++++++++++++++++++++++++++++--------------------------
 1 file changed, 30 insertions(+), 26 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index e702bac..22560f9 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -168,7 +168,7 @@ action potentially has more stationary points. We'll call $\Sigma$ the set of
     of $\mathcal S$, which produces the vertical reflection, and because in the
     pure 3-spin models $\mathcal S(-s)=-\mathcal S(s)$, which produces the
     simultaneous translation and inversion symmetry.
-  }
+  } \label{fig:example.action}
 \end{figure}
 
 Assuming $\mathcal S$ is holomorphic (and that the phase space $\Omega$ is
@@ -194,15 +194,15 @@ contour than our initial phase space $\Omega$.
 
   \caption{
     A schematic picture of the complex phase space for the circular $p$-spin
-    model and its standard integration contour. (Top, all): For real variables,
+    model and its standard integration contour. \textbf{Top:} For real variables,
     the model is a circle, and its analytic continuation is a kind of complex
-    hyperbola, here shown schematically in three dimensions. (Bottom, all):
+    hyperbola, here shown schematically in three dimensions. \textbf{Bottom:}
     Since the real manifold (the circle) is one-dimensional, the complex
     manifold has one complex dimension, here parameterized by the angle
-    $\theta$ on the circle. (Left): The integration contour over the real phase
-    space of the circular model. (Center): Complex analysis implies that the
+    $\theta$ on the circle. \textbf{Left:} The integration contour over the real phase
+    space of the circular model. \textbf{Center:} Complex analysis implies that the
     contour can be freely deformed without changing the value of the integral.
-    (Right): A funny deformation of the contour in which pieces have been
+    \textbf{Right:} A funny deformation of the contour in which pieces have been
     pinched off to infinity. So long as no poles have been crossed, even this
     is legal.
   }
@@ -261,17 +261,18 @@ group.
 
   \caption{
     A demonstration of the rules of thimble homology. Both figures depict the
-    complex-$\theta$ plane of an $N=2$ spherical $3$-spin model. The black
-    symbols lie on the stationary points of the action, and the grey regions
-    depict the sets $\tilde\Omega_T$ of well-behaved regions at infinity (here
-    $T=5$). (Left) Lines show the thimbles of each stationary point. The
+    complex-$\theta$ plane of action $\mathcal S$ featured in
+    Fig.~\ref{fig:example.action} with $\arg\beta=0.4$. The black symbols lie
+    on the stationary points of the action, and the grey regions depict the
+    sets $\tilde\Omega_T$ of well-behaved regions at infinity (here $T=5$).
+    \textbf{Left:} Lines show the thimbles of each stationary point. The
     thimbles necessary to recreate the cyclic path from left to right are
     darkly shaded, while those unnecessary for the task are lightly shaded.
-    Notice that all thimbles come and go from the well-behaved regions. (Right)
-    Lines show the antithimbles of each stationary point. Notice that those of
-    the stationary points involved in the contour (shaded darkly) all intersect
-    the desired contour (the real axis), while those not involved do not
-    intersect it.
+    Notice that all thimbles come and go from the well-behaved regions.
+    \textbf{Right:} Lines show the antithimbles of each stationary point.
+    Notice that those of the stationary points involved in the contour (shaded
+    darkly) all intersect the desired contour (the real axis), while those not
+    involved do not intersect it.
   } \label{fig:thimble.homology}
 \end{figure}
 
@@ -311,17 +312,20 @@ behavior can be seen in Fig.~\ref{fig:1d.stokes}.
   \includegraphics{figs/thimble_stokes_3.pdf}
 
   \caption{
-    An example of a Stokes point. (Left) The collection of thimbles necessary
-    to progress around from left to right, highlighted in a darker color, is
-    the same as it was in Fig.~\ref{fig:thimble.homology}. (Center) The thimble
-    $\mathcal J_\blacklozenge$ intersects the stationary point $\blacktriangle$
-    and its thimble, leading to a situation where the contour is not easily
-    defined using thimbles. This is a Stokes point. (Right) The Stokes point
-    has passed, and the collection of thimbles necessary to produce the path
-    has changed: now $\mathcal J_\blacktriangle$ must be included. Notice that
-    in this figure, because of the symmetry of the pure models, the thimble
-    $\mathcal J_\blacksquare$ also experiences a Stokes point, but this does
-    not result in a change to the path involving that thimble.
+    An example of a Stokes point in the continuation of the phase space
+    integral involving the action $\mathcal S$ featured in
+    Fig.~\ref{fig:example.action}. \textbf{Left:} $\arg\beta=1.176$. The collection of
+    thimbles necessary to progress around from left to right, highlighted in a
+    darker color, is the same as it was in Fig.~\ref{fig:thimble.homology}.
+    \textbf{Center:} $\arg\beta=1.336$. The thimble $\mathcal J_\blacklozenge$
+    intersects the stationary point $\blacktriangle$ and its thimble, leading
+    to a situation where the contour is not easily defined using thimbles. This
+    is a Stokes point. \textbf{Right:} $\arg\beta=1.496$. The Stokes point has passed,
+    and the collection of thimbles necessary to produce the path has changed:
+    now $\mathcal J_\blacktriangle$ must be included. Notice that in this
+    figure, because of the symmetry of the pure models, the thimble $\mathcal
+    J_\blacksquare$ also experiences a Stokes point, but this does not result
+    in a change to the path involving that thimble.
   } \label{fig:1d.stokes}
 \end{figure}
 
-- 
cgit v1.2.3-70-g09d2

