From 2bc53c86660edff2e5c3af1edcf65462b61d9bb8 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 7 Jun 2023 16:27:47 +0200
Subject: Lots of wroting and figure work.

---
 2-point.tex | 58 ++++++++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 52 insertions(+), 6 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 2a37c11..6cf2f60 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -787,12 +787,11 @@ The population of stationary points that are most common at each energy have the
   E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0
 \end{equation}
 between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this most common value, the energy and stability of the most common neighbors at small $\Delta q$ are
-\begin{equation}
-  E_1=E_0+\frac12\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big)
-\end{equation}
-\begin{equation}
+\begin{align} \label{eq:expansion.E.1}
+  E_1=E_0+\frac12\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\
+    \label{eq:expansion.mu.1}
   \mu_1=\mu_0-\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big)
-\end{equation}
+\end{align}
 Therefore, whether the energy and stability of nearby points increases or
 decreases from that of the reference point depends only on whether the energy
 of the reference point is above or below that of the most common population at
@@ -817,7 +816,54 @@ Note that this expression is only true for $\mu=\mu_\mathrm m$. Therefore,
 among marginal minima, when $E_0$ is greater than the threshold one finds
 neighbors at arbitrarily close distance. When $E_0$ is less than the threshold,
 the complexity of nearby points is negative, and there is a desert where none
-are found.
+are found. The properties of the nearby states above the threshold can be
+further quantified. The most common points are still given by
+\eqref{eq:expansion.E.1} and \eqref{eq:expansion.mu.1}, but the range of
+available points can also be computed, and one finds that the stability lies in
+the range
+$\mu_1=\mu_0+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$
+where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1}
+and
+\begin{equation}
+  \delta\mu_2=\frac{f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2}{f'(1)f''(1)^{3/4}}\sqrt{
+    \frac{E_0-E_\mathrm{th}}2\frac{f'(1)\big(f'''(1)-2f''(1)\big)+2f''(1)^2}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}
+  }
+\end{equation}
+Similarly, one finds that the energy lies in the range $E_1=E_0+\delta E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given by the coefficient in \eqref{eq:expansion.E.1} and
+\begin{equation}
+  \delta E_2
+  =\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\sqrt{
+    \frac{
+      \big[f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2\big]\big[
+        f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)
+      \big]
+      \big[
+        f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))
+      \big]
+    }
+    {3\big(f(1)(f'(1)+f''(1))-f'(1)^2\big)}
+  }
+\end{equation}
+and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic equation
+\begin{equation}
+  0=((16-(6-\delta q_0)\delta q_0)\delta q_0-12)f'(1)f'''(1)-2\delta q_0(f''(1)-f'(1))f''(1)
+\end{equation}
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/expansion_energy.pdf}
+  \hspace{1em}
+  \includegraphics{figs/expansion_stability.pdf}
+
+  \caption{
+    Demonstration of the convergence of the $(1-q)$-expansion for marginal
+    reference minima. Solid lines and shaded region show are the same as in
+    Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$.
+    The dotted lines show the expansion of most common neighbors, while the
+    dashed lines in both plots show the expansion for the minimum and maximum
+    energies and stabilities found at given $q$.
+  } \label{fig:expansion}
+\end{figure}
 
 
 \section{Isolated eigenvalue}
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