From 5e5b89fbab8298249d55b2d6413a8f501e0dbae2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 9 Jun 2023 16:16:30 +0200
Subject: Lots of work.

---
 2-point.tex | 169 +++++++++++++++++++++++++++++++++++++++++++++++-------------
 1 file changed, 133 insertions(+), 36 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 4e14f4c..bdcc8c8 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -101,7 +101,7 @@
   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, and invalidate pictures of
-  the arrangement of marginal states into a continuous `manifold.'
+  the arrangement most of marginal states into a continuous `manifold.'
 \end{abstract}
 
 \tableofcontents
@@ -118,14 +118,15 @@ However, in most models there is no quantitative correspondence between these
 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
+glasses: in the pure spherical models, the dynamic transition corresponds with
+the energy level at which marginal states dominate the free energy
 \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}
+the dynamical transition from any starting temperature
+\cite{Biroli_1999_Dynamical}. All of this can be understood in terms of the
+landscape structure.
 
 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
@@ -151,8 +152,11 @@ we focus on the neighborhoods of the marginal minima, to see if there is
 anything interesting to differentiate sets of them from each other. Though we
 find rich structure in this population, their properties pivot around the
 debunked threshold energy, and the apparent attractors of long-time dynamics
-are not distinguished by this measure. Therefore, with respect to the problem
-of dynamics this paper merely deepens the outstanding problems.
+are not distinguished by this measure. Moreover, we show that the usual picture of a
+marginal `manifold' of states separated by subextensive barriers is only true
+at the threshold energy, while at other energies marginal minima are far apart
+and separated by extensive barriers \cite{Kurchan_1996_Phase}. Therefore, with respect to the problem of
+dynamics this paper merely deepens the outstanding problems.
 
 In \S\ref{sec:model}, we introduce the mixed spherical models and discuss their
 properties. In \S\ref{sec:results}, we share the main results of the paper. In
@@ -258,7 +262,7 @@ marginal minima.
     eigenvalue outside a positive bulk spectrum is negative, destabilizing what
     would otherwise have been a stable minimum, producing an \emph{oriented
     index-one saddle}.
-  }
+  } \label{fig:spectra}
 \end{figure}
 
 In the pure spherical models, $E$ and $\mu$ cannot be fixed separately: fixing
@@ -267,17 +271,6 @@ of these models: marginal minima exist \emph{only} at one energy level, and
 therefore only that energy has the possibility of trapping the long-time
 dynamics.
 
-\subsection{Models of focus}
-
-In this study, we focus exclusively on models whose complexity is replica symmetric. We study two models of interest, both with concave $f''(q)^{-1/2}$: a $3+4$ model whose dynamics were studied extensively in \cite{Folena_2020_Rethinking}, given by
-\begin{equation}
-  f_{3+4}(q)=\frac12\big(q^3+q^4\big)
-\end{equation}
-and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics recently studied in \cite{Folena_2023_On} given by
-\begin{equation}
-  f_{3+8}(q)=\frac12\big(\tfrac{76}{100}q^3+\tfrac{24}{100}q^8\big)
-\end{equation}
-
 \begin{figure}
   \centering
   \includegraphics{figs/single_complexity.pdf}
@@ -294,6 +287,15 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r
   } \label{fig:complexities}
 \end{figure}
 
+In this study, we focus exclusively on the model studied in
+\cite{Folena_2020_Rethinking}, whose covariance function is given by
+\begin{equation}
+  f_{3+4}(q)=\frac12\big(q^3+q^4\big)
+\end{equation}
+First, it has concave $f''(q)^{-1/2}$, so at least the ground state complexity
+must be replica symmetric, and second, properties of its long-time dynamics
+have been extensively studied.
+
 \section{Results}
 \label{sec:results}
 
@@ -317,18 +319,37 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r
     set of stationary points than shown in the left plot. On both plots, the
     shading of the righthand part depicts the state of an isolated eigenvalue
     in the spectrum of the Hessian of the neighboring points. Those more
-    lightly shaded are minima with an isolated eigenvalue that does not change
-    their stability, i.e., $\lambda_\mathrm{min}>0$. Those more darkly shaded
-    are saddles with an isolated eigenvalue, either with many unstable
-    directions ($\mu_1<\mu_\mathrm m$) or with only one, corresponding to minima
-    destabilized by the isolated eigenvalue ($\mu_1>\mu_\mathrm m$). The
-    dot-dashed lines on both plots depict the trajectory of the solid line on
-    the other plot. In this case, the points lying nearest to the reference
-    minimum are saddles with $\mu<\mu_\mathrm m$, but with energies smaller than
-    the threshold energy.
+    lightly shaded are points with an isolated eigenvalue that does not change
+    their stability, e.g., corresponding with Fig.~\ref{fig:spectra}(d-e). The more
+    darkly shaded are oriented index-one saddles, e.g., corresponding with
+    Fig.~\ref{fig:spectra}(f). The dot-dashed lines on both plots depict the
+    trajectory of the solid line on the other plot. In this case, the points
+    lying nearest to the reference minimum are saddles with $\mu<\mu_\mathrm
+    m$, but with energies smaller than the threshold energy.
   } \label{fig:min.neighborhood}
 \end{figure}
 
+If the reference configuration is restricted to stable minima, then there is a
+gap in the overlap between those minima and their nearest neighbors in
+configuration space. We con characterize these neighbors as a function of their
+overlap and stability, with one example seen in
+Fig.~\ref{fig:min.neighborhood}. For stable minima, the qualitative results for
+the pure $p$-spin model continue to hold, with some small modifications
+\cite{Ros_2019_Complexity}.
+
+First, the nearest neighbor points are always oriented saddles, sometimes
+extensive saddles and sometimes index-one saddles (Fig.~\ref{fig:spectra}(d,
+f)). Like in the pure models, the emergence of oriented index-one saddles along
+the line of lowest-energy states at a given overlap occurs at the local minimum
+of this line. Unlike the pure models, neighbors exist for independent $\mu_1$
+and $E_1$, and the line of lowest-energy states at a given overlap is different
+from the line of maximally-stable states at a given overlap.
+
+Also like the pure models, there is a correspondence between the maximum of the
+zero-temperature Franz--Parisi potential restricted to minima of the specified
+type and the local maximum of the neighbor complexity along the line of
+lowest-energy states. This is seen in Fig.~\ref{fig:franz-parisi}.
+
 \begin{figure}
   \centering
   \includegraphics{figs/franz_parisi.pdf}
@@ -358,13 +379,13 @@ energy.
     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
+    stationary points at arbitrarily close distance, with a quadratic 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
+    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.
@@ -378,6 +399,17 @@ 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.
 
+We must put a small caveat here: in \emph{any} situation, this calculation
+admits order-one other marginal minima to lie a subextensive distance from the
+reference point. For such a population of points, $\Sigma_{12}=0$ and $q=1$,
+which is always a permitted solution when at least one marginal direction
+exists. These points are of course separated by small barriers from one
+another, but they also cover a vanishing piece of configuration space, and each
+such cluster of points is isolated by extensive barriers from each other
+cluster in the way described above. To move on a `manifold' nearby marginal
+minima within such a cluster cannot describe aging, since the overlap with the
+initial condition will never change from one.
+
 \begin{figure}
   \includegraphics{figs/nearest_energies_above.pdf}
   \includegraphics{figs/nearest_stabilities_above.pdf}
@@ -419,12 +451,46 @@ marginal minima are isolated from each other.
   } \label{fig:marginal.prop.above}
 \end{figure}
 
+\begin{figure}
+  \includegraphics{figs/nearest_energies_thres.pdf}
+  \includegraphics{figs/nearest_stabilities_thres.pdf}
+  \includegraphics{figs/nearest_marginal_thres.pdf}
+
+  \caption{
+    The neighborhood of marginal states at the threshold energy $E_0=E_\mathrm{th}$.
+  } \label{fig:marginal.prop.thres}
+\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
 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.
+energy must  be reached after an extensive distance in
+configuration space without encountering any stationary point. A version of
+this story was told a long time ago by the authors of
+\cite{Kurchan_1996_Phase}, who write on aging in the pure spherical models
+where the limit of $N\to\infty$ is taken before that of $t\to\infty$: ``it is
+important to remark that this [...]\ does \emph{not} mean that the system
+relaxes into a near-threshold state: at all finite times an infinite system has
+a Hessian with an \emph{infinite} number of directions in which the energy is a
+maximum. [...] We have seen that the saddles separating threshold minima are
+typically $O(N^{1/3})$ above the threshold level, while the energy is at all
+finite times $O(N)$ above this level.'' In the present case of the mixed
+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
@@ -442,7 +508,9 @@ configuration space without encountering any stationary point.
 
 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
+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
@@ -471,7 +539,8 @@ measure conditioned on the energy density $E$ and stability $\mu$ of the points,
 While $\mu$ is strictly the trace of the Hessian, we call it the stability
 because in this family of models all stationary points have a bulk spectrum of
 the same shape, shifted by different constants. The stability $\mu$ sets this
-shift, and therefore determines if the spectrum has bulk support on zero.
+shift, and therefore determines if the spectrum has bulk support on zero. See
+Fig.~\ref{fig:spectra} for examples.
 
 We want the typical number of stationary points with energy density
 $E_1$ and stability $\mu_1$ that lie a fixed overlap $q$ from a reference
@@ -1421,9 +1490,31 @@ directly related to $x_0$. This tangent vector is $\mathbf x_{0\leftarrow
 1}=\frac1{1-q}\big(\pmb\sigma_0-q\mathbf s_a\big)$, which is normalized and
 lies strictly in the tangent plane of $\mathbf s_a$. Then
 \begin{equation}
-  \frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N
+  q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N
   =\frac{x_0}{1-q}
 \end{equation}
+The emergence of an isolated eigenvalue and its associated eigenvector are
+shown in Fig.~\ref{fig:isolated.eigenvalue}, for the same reference point
+properties as in Fig.~\ref{fig:min.neighborhood}.
+
+\begin{figure}
+  \includegraphics{figs/isolated_eigenvalue.pdf}
+  \hfill
+  \includegraphics{figs/eigenvector_overlap.pdf}
+
+  \caption{
+    Properties of the isolated eigenvalue and the overlap of its associated
+    eigenvector with the direction of the reference point. These curves
+    correspond with the lower solid curve in Fig.~\ref{fig:min.neighborhood}.
+    \textbf{Left:} The value of the minimum eigenvalue as a function of
+    overlap. The dashed line shows the continuation of the bottom of the
+    semicircle. Where the dashed line separates from the solid line, the
+    isolated eigenvalue has appeared. \textbf{Right:} The overlap between the
+    eigenvector associated with the minimum eigenvalue and the direction of the
+    reference point. The overlap is zero until an isolated eigenvalue appears,
+    and then it grows continuously until the nearest neighbor is reached.
+  } \label{fig:isolated.eigenvalue}
+\end{figure}
 
 \section{Franz--Parisi potential}
 \label{sec:franz-parisi}
@@ -1507,8 +1598,14 @@ $z$, $y_1$, and $q_0$.
 \section{Conclusion}
 \label{sec:conclusion}
 
+We have computed the complexity of neighboring stationary points for the mixed
+spherical models. When we studied the neighborhoods of marginal minima, we
+found something striking: only those at the threshold energy have other
+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.
+generalized to landscapes with replica symmetry broken complexities \cite{Kent-Dobias_2022_How}.
 
 \paragraph{Acknowledgements}
 
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