From 924b94d4d4da0dd0c919892fa98fd93c8c13e011 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 6 Jun 2023 13:49:51 +0200
Subject: Some writing, lots of figure work.

---
 2-point.tex | 12 ++++++++++++
 1 file changed, 12 insertions(+)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index efc7a71..f8d700a 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -743,6 +743,18 @@ intuitive: stable minima have an effective repulsion between points, and one
 always finds a sufficiently small $\Delta q$ that no stationary points are
 point any nearer. For the marginal minima, it is not clear that the same should be true.
 
+When $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion is
+\begin{equation}
+  \Sigma_{12}
+  =\frac12\frac{f'''(1)\big(f'(1)(f''(1)+f'''(1))-f''(1)^2\big)}{f''(1)^{3/2}\big(f(1)(f'(1)+f''(1))-f'(1)^2\big)}
+  \left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)\big(E_0-E_\textrm{th}\big)(1-q)^2+O\big((1-q)^3\big)
+\end{equation}
+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.
+
 
 \section{Isolated eigenvalue}
 \label{sec:eigenvalue}
-- 
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