Broke of much of the isolated eigenvalue to an appendix.

1 files changed, 109 insertions, 105 deletions

diff --git a/2-point.tex b/2-point.tex
index 238726b..9be13e9 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -1218,6 +1218,115 @@ $\pmb\sigma$ replicas constrained to lie at fixed overlap with \emph{all} the
 $\mathbf s$ replicas, and the second is the only of the $\mathbf s$ replicas at
 which the Hessian is evaluated.
 
+
+In this solution, we simultaneously find the smallest eigenvalue and information
+about the orientation of its associated eigenvector: namely, its overlap with
+the tangent vector that points directly toward the reference spin. This is
+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}
+  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{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.
+
+This has implications for pictures of relaxation and aging. In most $p+s$
+models studied, quenches from infinite to zero temperature (gradient descent
+starting from a random point) relax towards marginal states with energies above
+the threshold energy \cite{Folena_2023_On}, while at least in some models a
+quench to zero temperature from a temperature around the dynamic transition
+relaxes towards marginal states with energies below the threshold energy
+\cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially
+Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the
+neighborhoods of marginal states above and below the threshold are quite
+different, and yet the emergent aging behaviors relaxing toward states above and
+below the threshold seem to be the same. Therefore, this kind of dynamics
+appears to be insensitive to the neighborhood of the marginal state being
+approached. To understand something better about why certain states attract the
+dynamics in certain situations, nonlocal information, like the
+structure of their entire basin of attraction, seems vital.
+
+It is possible that replica symmetry breaking among the constrained stationary
+points could change the details of the two-point complexity of very nearby
+states. Indeed, it is difficult to rule out \textsc{rsb} in complexity
+calculations. However, such corrections would not change the overarching
+conclusions of this paper, namely that most marginal minima are separated from
+each other by a macroscopic overlap gap and high barriers. This is because the
+replica symmetric complexity bounds any \textsc{rsb} complexities from above,
+and so \textsc{rsb} corrections can only decrease the complexity. Therefore,
+the overlap gaps, which correspond to regions of negative complexity, cannot be
+removed by a more detailed saddle point ansatz.
+
+Our calculation studied the neighborhood of typical reference points with the
+given energy and stability. However, it is possible that marginal minima with
+atypical neighborhoods actually attract the dynamics. To determine this, a
+different type of calculation is needed. As our calculation is akin to the
+quenched Franz--Parisi potential, study of atypical neighborhoods would entail
+something like the annealed Franz--Parisi approach, i.e.,
+\begin{equation}
+  \Sigma^*(E_0,\mu_0,E_1,\mu_1,q)=\frac1N\overline{\log\left(
+      \int d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)\,d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)
+  \right)}
+\end{equation}
+which puts the two points on equal footing. This calculation and exploration of
+the atypical neighborhoods it reveals is a clear future direction.
+
+The methods developed in this paper are straightforwardly (if not easily)
+generalized to landscapes with replica symmetry broken complexities
+\cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features
+of this study would persist, with neighboring states being divided into
+different clusters based on the \textsc{rsb} order but with the basic presence
+or absence of overlap gaps and the nature of the stability of near-neighbors
+remaining unchanged. Interesting structure might emerge in the arrangement of
+marginal states in \textsc{frsb} systems, where the ground state itself is
+marginal and coincides with the threshold.
+
+\paragraph{Acknowledgements}
+
+The author would like to thank Valentina Ros, Giampaolo Folena, Chiara
+Cammarota, and Jorge Kurchan for useful discussions related to this work.
+
+\paragraph{Funding information}
+
+JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the
+INFN.
+
+\appendix
+
+\section{Details of calculation for the isolated eigenvalue}
+\label{sec:eigenvalue-details}
+
 Using the same methodology as above, the disorder-dependent terms are captured in the linear operator
 \begin{equation}
   \mathcal O(\mathbf t)=
@@ -1588,111 +1697,6 @@ $y^2\geq f''(1)$. In practice, there is at most \emph{one} $y$ which produces a
 zero eigenvalue of $B-yC$ and satisfies this inequality, so the solution seems
 to be unique.
 
-In this solution, we simultaneously find the smallest eigenvalue and information
-about the orientation of its associated eigenvector: namely, its overlap with
-the tangent vector that points directly toward the reference spin. This is
-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}
-  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{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.
-
-This has implications for pictures of relaxation and aging. In most $p+s$
-models studied, quenches from infinite to zero temperature (gradient descent
-starting from a random point) relax towards marginal states with energies above
-the threshold energy \cite{Folena_2023_On}, while at least in some models a
-quench to zero temperature from a temperature around the dynamic transition
-relaxes towards marginal states with energies below the threshold energy
-\cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially
-Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the
-neighborhoods of marginal states above and below the threshold are quite
-different, and yet the emergent aging behaviors relaxing toward states above and
-below the threshold seem to be the same. Therefore, this kind of dynamics
-appears to be insensitive to the neighborhood of the marginal state being
-approached. To understand something better about why certain states attract the
-dynamics in certain situations, nonlocal information, like the
-structure of their entire basin of attraction, seems vital.
-
-It is possible that replica symmetry breaking among the constrained stationary
-points could change the details of the two-point complexity of very nearby
-states. Indeed, it is difficult to rule out \textsc{rsb} in complexity
-calculations. However, such corrections would not change the overarching
-conclusions of this paper, namely that most marginal minima are separated from
-each other by a macroscopic overlap gap and high barriers. This is because the
-replica symmetric complexity bounds any \textsc{rsb} complexities from above,
-and so \textsc{rsb} corrections can only decrease the complexity. Therefore,
-the overlap gaps, which correspond to regions of negative complexity, cannot be
-removed by a more detailed saddle point ansatz.
-
-Our calculation studied the neighborhood of typical reference points with the
-given energy and stability. However, it is possible that marginal minima with
-atypical neighborhoods actually attract the dynamics. To determine this, a
-different type of calculation is needed. As our calculation is akin to the
-quenched Franz--Parisi potential, study of atypical neighborhoods would entail
-something like the annealed Franz--Parisi approach, i.e.,
-\begin{equation}
-  \Sigma^*(E_0,\mu_0,E_1,\mu_1,q)=\frac1N\overline{\log\left(
-      \int d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)\,d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)
-  \right)}
-\end{equation}
-which puts the two points on equal footing. This calculation and exploration of
-the atypical neighborhoods it reveals is a clear future direction.
-
-The methods developed in this paper are straightforwardly (if not easily)
-generalized to landscapes with replica symmetry broken complexities
-\cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features
-of this study would persist, with neighboring states being divided into
-different clusters based on the \textsc{rsb} order but with the basic presence
-or absence of overlap gaps and the nature of the stability of near-neighbors
-remaining unchanged. Interesting structure might emerge in the arrangement of
-marginal states in \textsc{frsb} systems, where the ground state itself is
-marginal and coincides with the threshold.
-
-\paragraph{Acknowledgements}
-
-The author would like to thank Valentina Ros, Giampaolo Folena, Chiara
-Cammarota, and Jorge Kurchan for useful discussions related to this work.
-
-\paragraph{Funding information}
-
-JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the
-INFN.
-
-\appendix
-
 \section{Franz--Parisi potential}
 \label{sec:franz-parisi}