From e10bf44c5261d7f025ffa9e88e0bef4b2783e7d3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 26 Aug 2024 10:28:04 +0200
Subject: New figures and removing old figures.

---
 topology.tex | 23 +++++++++++++++++++++++
 1 file changed, 23 insertions(+)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 4c14207..102dc6b 100644
--- a/topology.tex
+++ b/topology.tex
@@ -140,6 +140,29 @@ putting strong constraints on the resulting topology and geometry.
 
 \subsection{Topology of solutions to many equations and the satisfiability transition}
 
+\begin{figure}
+  \includegraphics{figs/action_1.pdf}
+  \hspace{-3.5em}
+  \includegraphics{figs/action_3.pdf}
+
+  \caption{
+    The effective action governing the as a function of the overlap
+    $m=\frac1N\mathbf x\cdot\mathbf x_0$ with the height direction for two
+    different homogeneous polynomial functions and a variety of $V_0$. In both
+    plots $\alpha=\frac12$. \textbf{Left:} With linear functions there are two
+    regimes. For small $V_0$, there are maxima at $m=\pm m^*$ where the action
+    is zero, while after the satisfiability transition at $V_0=V_\text{\textsc{sat}}=1$, $m^*$
+    goes to zero and the action becomes negative. \textbf{Left:} With nonlinear
+    functions there are four regimes. For small $V_0$ the behavior is the same
+    as in the linear case, with zero action. After an onset transition at
+    $V_0=V_\text{on}\simeq1.099$ the maxima are at the edge of validity of the
+    action and the action is positive. At a shattering transition at
+    $V_0=V_\text{sh}\simeq1.394$, $m^*$ goes to zero and the action is positive.
+    Finally, at the satisfiability transition
+    $V_0=V_\text{\textsc{sat}}\simeq1.440$ the action becomes negative.
+  }
+\end{figure}
+
 \begin{figure}
   \includegraphics[width=0.245\textwidth]{figs/connected.pdf}
   \includegraphics[width=0.245\textwidth]{figs/coexist.pdf}
-- 
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