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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}