From 7671ec1a60f13693287c594a620a099d09c0c463 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 9 Jun 2023 16:34:48 +0200
Subject: Some changes.

---
 2-point.tex | 64 ++++++++++++++++++-------------------------------------------
 1 file changed, 19 insertions(+), 45 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index bdcc8c8..bb7128a 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -375,7 +375,7 @@ energy.
 \begin{itemize}
   \item \textbf{Energies below the threshold.} Marginal states have a
     macroscopic gap in their overlap with nearby minima and saddles. The
-    nearest stationary points are barriers with a single downward direction,
+    nearest stationary points are saddles with a single downward direction,
     and always have a higher energy density than the reference state.
 
   \item \textbf{Energies above the threshold.} Marginal states have neighboring
@@ -386,9 +386,9 @@ energy.
 
   \item \textbf{At the threshold energy.} Marginal states have neighboring
     stationary points at arbitrarily close distance, with a cubic pseudogap
-    in their complexity. The nearest ones are saddle points with an extensive
-    number of downward directions, save for a single downward direction, and
-    can be found at the same energy density as the reference state.
+    in their complexity. The nearest ones are oriented saddle points with an
+    extensive number of downward directions and can be found at the same energy
+    density as the reference state.
 \end{itemize}
 
 This leads us to some general conclusions. First, at all energy densities
@@ -446,7 +446,7 @@ initial condition will never change from one.
     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
+    eigenvalue appears. Marginal states below the threshold are always
     separated by a gap in their overlap.
   } \label{fig:marginal.prop.above}
 \end{figure}
@@ -457,7 +457,16 @@ initial condition will never change from one.
   \includegraphics{figs/nearest_marginal_thres.pdf}
 
   \caption{
-    The neighborhood of marginal states at the threshold energy $E_0=E_\mathrm{th}$.
+    The neighborhood of marginal minima at the threshold energy
+    $E_0=E_\mathrm{th}$. In all plots, the dashed lines show the population of
+    most common neighbors at the given overlap $q$.
+    \textbf{Left:} The range of energies $E_1$ at which nearby points are
+    found. The approach of both the minimum and maximum energies goes like
+    $(1-q)^3$. \textbf{Center:} The range of stabilities $\mu_1$ at which nearby
+    points are found. The approach of both limits goes like $(1-q)^2$.
+    \textbf{Right:} The range of nearby marginal minima. The more darkly shaded
+    region denotes where an isolated eigenvalue appears. Marginal minima at the
+    threshold lie asymptotically close together.
   } \label{fig:marginal.prop.thres}
 \end{figure}
 
@@ -480,44 +489,6 @@ spherical models, where \cite{Folena_2020_Rethinking} has shown aging dynamics
 asymptotically approaching marginal states that we have shown have $O(N)$
 saddles separating them, this lesson must be taken all the more seriously.
 
-There is one feature of marginal minima under the threshold energy worthy of
-special note, perhaps relevant to the problems of the asymptotic dynamics
-opened by \cite{Folena_2020_Rethinking}. Fig.~\ref{fig:dom.marg.below} shows
-the energy of the most common set of marginal minima at overlap $q$ from a
-reference marginal minimum of varying energy $E_0$. The figure also indicates
-where an isolated eigenvalue exists or does not. Since the spectrum is marginal
-for the neighboring states in question, any isolated eigenvalue would transform
-them into index-one saddles. At some energy
-$E_0=E_\textrm{cross}=-1.699\,619\,496\ldots$, the dominant points at the
-threshold energy transition from being oriented index-one saddles into marginal
-minima.
-
-\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 $o(N)$ barriers exactly at the threshold energy. Perhaps some interaction with or above this manifold is necessary for the emergence of aging dynamics, and once equilibrated below it exponential convergence to the inherent state is inevitable. From this reasoning, $E_\mathrm{cross}$ is the highest energy of marginal minima below the threshold such that
-
-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}
 
@@ -1605,10 +1576,13 @@ marginal minima nearby. For the many marginal minima away from the threshold
 (including the exponential majority), there is a gap in overlap between them.
 
 The methods developed in this paper are straightforwardly (if not easily)
-generalized to landscapes with replica symmetry broken complexities \cite{Kent-Dobias_2022_How}.
+generalized to landscapes with replica symmetry broken complexities \cite{Kent-Dobias_2023_How}.
 
 \paragraph{Acknowledgements}
 
+The author would like to thank Valentina Ros, Giampaolo Folena, and Chiara
+Cammarota for useful discussions related to this work.
+
 \paragraph{Funding information}
 
 \printbibliography
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