From f0df8efd0a5165823f888dc6c812351a612d2bbd Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 6 Jun 2023 19:09:05 +0200
Subject: Lots of writing and figures.

---
 2-point.tex | 64 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 64 insertions(+)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index f8d700a..2a37c11 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -339,6 +339,46 @@ being connected by subextensive energy barriers can only describe the
 collection of marginal minima at the threshold energy. Both below and above,
 marginal minima are isolated from each other.
 
+\begin{figure}
+  \includegraphics{figs/nearest_energies_above.pdf}
+  \includegraphics{figs/nearest_stabilities_above.pdf}
+  \includegraphics{figs/nearest_marginal_above.pdf}
+
+  \caption{
+    The neighborhood of marginal states at several energies above the threshold
+    energy. \textbf{Left:} The range of energies $E_1$ at which nearby states
+    are found. For any $E_0>E_\mathrm{th}$, there always exists a $q$
+    sufficiently close to one such that the nearby states have strictly greater
+    energy than the reference state. \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.
+  } \label{fig:marginal.prop.above}
+\end{figure}
+
+\begin{figure}
+  \includegraphics{figs/nearest_energies_below.pdf}
+  \includegraphics{figs/nearest_stabilities_below.pdf}
+  \includegraphics{figs/nearest_marginal_below.pdf}
+
+  \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.
+  } \label{fig:marginal.prop.above}
+\end{figure}
+
 This has implications for how quench dynamics should be interpreted. When
 marginal states are approached above the threshold energy, they must have been
 via the neighborhood of saddles with an extensive index, not other marginal
@@ -346,6 +386,30 @@ states. On the other hand, marginal states approached below the threshold
 energy must, in the end, be reached after an extensive distance in
 configuration space without encountering any stationary point.
 
+\begin{figure}
+  \centering
+  \includegraphics{figs/dom_marg_below.pdf}
+
+  \caption{
+    The energy $E_1$ of the most common set of marginal states at overlap $q$
+    with a marginal reference state with energy $E_0$. The lines are solid when
+    the constrained states are marginal \emph{minima}, and dashed when they are
+    saddles with an isolated eigenvalue. There is some energy of the reference
+    state $E_0=E_\text{cross}=-1.699\,619\,496\ldots$ where this transition
+    crosses the threshold energy.
+  } \label{fig:dom.marg.below}
+\end{figure}
+
+The difference between this crossing energy and the state following energy
+estimated in \cite{Folena_2020_Rethinking} is about $0.0002$, within the
+uncertainties reported in that work. Why should this point be relevant to the onset of state following dynamics? Consider the implications of what we have said about the marginal manifold, which only exists as a continuum of order-one barriers exactly at the threshold energy. The construction in that paper is not
+compatible with an interpretation of the state following temperature we have
+given. For instance, the value of $q_{12}\equiv q$ reported in that work is
+quite different from the $q=0.729\,981\,886\ldots$ that characterizes these
+states. To assess this shaky conjecture, study of the dynamics in other mixed
+spherical models would be most helpful, so as to compare with their
+complexities.
+
 \section{Complexity}
 \label{sec:complexity}
 
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