From 30dcb206d981da6d3e33e107da2913190dee9aec Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 9 Jun 2023 09:34:42 +0200
Subject: Some writing.

---
 2-point.tex | 19 ++++++++++---------
 1 file changed, 10 insertions(+), 9 deletions(-)

diff --git a/2-point.tex b/2-point.tex
index 7c7d345..4e14f4c 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -406,15 +406,16 @@ marginal minima are isolated from each other.
   \caption{
     The neighborhood of marginal states at several energies below the threshold
     energy. \textbf{Left:} The range of energies $E_1$ at which nearby states
-    are found. For any $E_0<E_\mathrm{th}$, the nearest class of states is at a
-    nonzero distance, and their energies are higher than that of the reference
-    configuration. \textbf{Center:} The range of stabilities $\mu_1$ at which
-    nearby states are found. As below, there is always a sufficiently large
-    overlap beyond which all nearby states are saddle with an extensive number
-    of downward directions. \textbf{Right:} The range of energies at which
-    \emph{other} marginal states are found. Here, the more darkly shaded
-    regions denote where an isolated eigenvalue appears. Marginal states above
-    the threshold are always separated by a gap in their overlap.
+    are found. For any $E_0<E_\mathrm{th}$, the nearest class of states is at
+    an extensive distance, and their energies are higher than that of the
+    reference configuration. \textbf{Center:} The range of stabilities $\mu_1$
+    at which nearby states are found. For $E_0$ near the threshold, the nearest
+    states are always index-one saddles with $\mu>\mu_\mathrm m$, but as the
+    overlap gap widens their population becomes model-dependent.
+    \textbf{Right:} The range of energies at which \emph{other} marginal states
+    are found. Here, the more darkly shaded regions denote where an isolated
+    eigenvalue appears. Marginal states belo the threshold are always
+    separated by a gap in their overlap.
   } \label{fig:marginal.prop.above}
 \end{figure}
 
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