From 01cefa889e8f5e6d3034360704edf92fd9cc8060 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 5 Jun 2023 18:15:08 +0200
Subject: Lots of work.

---
 2-point.tex | 73 ++++++++++++++++++++++++++++++++++++++++++++++++-------------
 1 file changed, 58 insertions(+), 15 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 9236cf9..bf1a5ca 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -100,10 +100,8 @@
   eigenvalue in their spectrum due to their proximity. Despite finding rich
   structure not present in the pure models, we find nothing that distinguishes
   the points that do attract the dynamics. Instead, we find new geometric
-  significance of the old threshold energy.
-
-  Found significance of the state following energy! Energy at which population
-  of nearest marginal states with energies below the threshold have no isolated eigenvalue.
+  significance of the old threshold energy, and invalidate pictures of
+  the arrangement of marginal states into a continuous `manifold.'
 \end{abstract}
 
 \tableofcontents
@@ -117,17 +115,17 @@ spin glasses, certain inference and optimization problems, and more
 an energy or cost landscape, whether due to the proliferation of metastable
 states, to the raising of barriers which cause effective dynamic constraints.
 However, in most models there is no quantitative correspondence between these
-properties and the behavior.
-
-In perhaps the simplest mean-field model of glasses, such a correspondence
-exists and is well understood. In the pure spherical models, the dynamic
-transition corresponds precisely with the energy level at which all marginal
-minima are concentrated. At that level, called the \emph{threshold energy}
-$E_\mathrm{th}$, slices of the landscape at fixed energy undergo a percolation
-transition. In fact, this threshold energy is significant in other ways: it
-attracts the long-time dynamics after quenches in temperature to below the
-dynamical transition from any starting temperature. All of this can be understood in terms of the landscape structure.
-\cite{Biroli_1999_Dynamical}
+properties and the behavior they are purported to describe.
+
+There is such a correspondence in one of the simplest mean-field model of
+glasses: in the pure spherical models, the dynamic transition corresponds
+precisely with the energy level at which all marginal minima are concentrated.
+At that level, called the \emph{threshold energy} $E_\mathrm{th}$, slices of
+the landscape at fixed energy undergo a percolation transition. In fact, this
+threshold energy is significant in other ways: it attracts the long-time
+dynamics after quenches in temperature to below the dynamical transition from
+any starting temperature. All of this can be understood in terms of the
+landscape structure. \cite{Biroli_1999_Dynamical}
 
 In slightly less simple models, the mixed spherical models, the story changes.
 There are now a range of energies with exponentially many marginal minima. It
@@ -260,6 +258,8 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r
 \section{Results}
 \label{sec:results}
 
+\subsection{Barriers around deep states}
+
 \begin{figure}
   \includegraphics{figs/gapped_min_energy.pdf}
   \raisebox{5em}{\includegraphics{figs/gapped_min_energy_legend.pdf}}
@@ -303,6 +303,49 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r
   } \label{fig:franz-parisi}
 \end{figure}
 
+\subsection{Geometry of marginal states}
+
+The set of marginal states is of special interest. First, it has more structure
+than in the pure models, with different types of marginal states being found at
+different energies. Second, these states attract the dynamics, and so are the
+inevitable end-point of equilibrium and algorithmic processes. We find,
+surprisingly, that the properties of marginal states pivot around the threshold
+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,
+    and always have a higher energy density than the reference state.
+
+  \item \textbf{Energies above the threshold.} Marginal states have neighboring
+    stationary points at arbitrarily close distance, with a linear pseudogap in
+    their complexity. The nearest ones are \emph{strictly} saddle points with
+    an extensive number of downward directions and always have a higher energy
+    density than the reference state.
+
+  \item \textbf{At the threshold energy.} Marginal states have neighboring
+    stationary points at arbitrarily close distance, with a quadratic 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.
+\end{itemize}
+
+This leads us to some general conclusions. First, at all energy densities
+except at the threshold energy, \emph{marginal minima are separated by
+extensive energy barriers}. Therefore, the picture of a marginal
+\emph{manifold} of many (even all) marginal states lying arbitrarily close and
+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.
+
+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
+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.
+
 \section{Complexity}
 \label{sec:complexity}
 
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