From eaab0737ca427954587eb7aca26bea072d9abe65 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 26 Aug 2024 00:56:54 +0200
Subject: Some tweaks.

---
 topology.tex | 46 ++++++++++++++++++++++------------------------
 1 file changed, 22 insertions(+), 24 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index f420411..4d2d660 100644
--- a/topology.tex
+++ b/topology.tex
@@ -134,6 +134,28 @@ putting strong constraints on the resulting topology and geometry.
 
 \subsection{Topology of solutions to many equations and the satisfiability transition}
 
+\begin{figure}
+  \includegraphics[width=0.245\textwidth]{figs/connected.pdf}
+  \includegraphics[width=0.245\textwidth]{figs/coexist.pdf}
+  \includegraphics[width=0.245\textwidth]{figs/shattered.pdf}
+  \includegraphics[width=0.245\textwidth]{figs/gone.pdf}
+
+  \includegraphics{figs/bar.pdf}
+
+  \caption{
+    Cartoon of the topology of the solution manifold implied by our
+    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
+    function. For $V_0<V_\text{on}$, the manifold has a single connected
+    component. Above the onset with $V_\text{on}<V_0<V_\text{sh}$, the manifold
+    has a large connected component around the equator, and many disconnected
+    pieces in a certain range of latitudes. Above the shattering transition, or
+    $V_\text{sh}<V_0<V_\text{\textsc{sat}}$, the large connected
+    component vanishes and small disconnected pieces occupy the entire equatorial
+    region. Finally, above the satisfiability transition
+    $V_\text{\textsc{sat}}$ the manifold vanishes.
+  } \label{fig:cartoons}
+\end{figure}
+
 \begin{figure}
   \includegraphics{figs/phases_1.pdf}
   \hspace{-3em}
@@ -155,30 +177,6 @@ putting strong constraints on the resulting topology and geometry.
   }
 \end{figure}
 
-\begin{figure}
-  \includegraphics[width=0.24\textwidth]{figs/connected.pdf}
-  \hfill
-  \includegraphics[width=0.24\textwidth]{figs/coexist.pdf}
-  \hfill
-  \includegraphics[width=0.24\textwidth]{figs/shattered.pdf}
-  \hfill
-  \includegraphics[width=0.24\textwidth]{figs/gone.pdf}
-
-  \includegraphics{figs/bar.pdf}
-
-  \caption{
-    Cartoon of the topology of the solution manifold implied by our
-    calculation. The arrow shows the vector $\mathbf x_0$ defining the height
-    function. The region of solutions is marked in black, and the critical points
-    of the height function restricted to this region are marked with a point.
-    For $\alpha<1$, there are few simply connected regions with most of the
-    minima and maxima contributing to the Euler characteristic concentrated at
-    the height $m^*$. For $\alpha\geq1$, there are many simply
-    connected regions and most of their minima and maxima are concentrated at
-    the equator.
-  } \label{fig:cartoons}
-\end{figure}
-
 \subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold}
 
 \begin{figure}
-- 
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