From f63692f516c2b841ab86dd2e239cf158dbb657d1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 8 Jun 2023 23:21:02 +0200
Subject: Lots of work.

---
 2-point.tex | 69 +++++++++++++++++++++++++++++++++++++++++++++++--------------
 1 file changed, 53 insertions(+), 16 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 9d951e9..7c7d345 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -86,7 +86,7 @@
 }
 
 \author{Jaron Kent-Dobias}
-\affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma}
+\affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma I}
 
 \maketitle
 \begin{abstract}
@@ -113,19 +113,19 @@ over a short parameter interval. These include actual (structural) glasses,
 spin glasses, certain inference and optimization problems, and more
 \cite{lots}. Glassiness is qualitatively understood to arise from structure of
 an energy or cost landscape, whether due to the proliferation of metastable
-states, to the raising of barriers which cause effective dynamic constraints.
+states, or 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 they are purported to describe.
+landscape properties and the dynamic 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}
+precisely with the energy level at which all marginal minima are concentrated
+\cite{Castellani_2005_Spin-glass}. 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
@@ -226,6 +226,41 @@ semicircle and becomes negative, the stationary point is a minimum. Finally,
 when $\mu=\mu_\mathrm m$, the edge of the semicircle touches zero and we have
 marginal minima.
 
+\begin{figure}
+  \includegraphics{figs/spectrum_saddle.pdf}
+  \hfill
+  \includegraphics{figs/spectrum_marginal.pdf}
+  \hfill
+  \includegraphics{figs/spectrum_minimum.pdf}\\
+
+  \vspace{1em}
+
+  \includegraphics{figs/spectrum_saddle_2.pdf}
+  \hfill
+  \includegraphics{figs/spectrum_minimum_2.pdf}
+  \hfill
+  \includegraphics{figs/spectrum_saddle_3.pdf}
+
+  \caption{
+    Illustration of the interpretation of the stability $\mu$, which sets the
+    location of the center of the eigenvalue spectrum. In the top row we have
+    spectra without an isolated eigenvalue. \textbf{(a)} $\mu<\mu_\mathrm m$,
+    there are an extensive number of downward directions, and the associated
+    point is an \emph{extensive saddle}. \textbf{(b)} $\mu=\mu_\mathrm m$ and
+    we have a \emph{marginal minimum} with asymptotically flat directions.
+    \textbf{(c)} $\mu>\mu_\mathrm m$, all eigenvalues are positive, and the
+    point is a \emph{stable minimum}. On the bottom we show what happens in the
+    presence of an isolated eigenvalue. \textbf{(d)} One eigenvalue leaves the
+    bulk spectrum of a saddle point and it remains a saddle point, but now with
+    an eigenvector correlated with the orientation of the reference vector, so
+    we call this a \emph{oriented saddle}. \textbf{(e)} The same happens for
+    a minimum, and we can call it an \emph{oriented minimum}. \textbf{(f)} One
+    eigenvalue outside a positive bulk spectrum is negative, destabilizing what
+    would otherwise have been a stable minimum, producing an \emph{oriented
+    index-one saddle}.
+  }
+\end{figure}
+
 In the pure spherical models, $E$ and $\mu$ cannot be fixed separately: fixing
 one uniquely fixes the other. This property leads to the great simplification
 of these models: marginal minima exist \emph{only} at one energy level, and
@@ -247,13 +282,15 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r
   \centering
   \includegraphics{figs/single_complexity.pdf}
   \caption{
-    Plots of the complexity (logarithm of the number of stationary points) for
-    the mixed spherical models studied in this paper. Energies and stabilities
+    Plot of the complexity (logarithm of the number of stationary points) for
+    the $3+4$ mixed spherical model studied in this paper. Energies and stabilities
     of interest are marked, including the ground state energy and stability
-    $E_\mathrm{gs}$ and $\mu_\mathrm{gs}$, the marginal stability $\mu_\mathrm
-    m$, and the threshold energy $E_\mathrm{th}$. The line shows the location
-    of the most common type of stationary point at each energy level. Estimated
-    locations of notable attractors of the dynamics are highlighted.
+    $E_\mathrm{gs}$, the marginal stability $\mu_\mathrm
+    m$, and the threshold energy $E_\mathrm{th}$. The blue line shows the location
+    of the most common type of stationary point at each energy level. The
+    highlighted red region shows the approximate range of minima which attract
+    aging dynamics from a quench to zero temperature found in
+    \cite{Folena_2020_Rethinking}.
   } \label{fig:complexities}
 \end{figure}
 
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