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diff --git a/2-point.tex b/2-point.tex index acbba87..238726b 100644 --- a/2-point.tex +++ b/2-point.tex @@ -409,23 +409,6 @@ 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} - - \caption{ - Comparison of the lowest-energy stationary points at overlap $q$ with a - reference minimum of $E_0=-1.71865<E_\mathrm{th}$ and - $\mu_0=6.1>\mu_\mathrm m$ (yellow, top), and the zero-temperature Franz--Parisi potential - with respect to the same reference minimum (blue, bottom). The two curves - coincide precisely at their minimum $q=0$ and at the local maximum $q\simeq0.5909$. - } \label{fig:franz-parisi} -\end{figure} \subsection{Geometry of marginal states} @@ -1638,6 +1621,78 @@ properties as in Fig.~\ref{fig:min.neighborhood}. } \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} @@ -1734,75 +1789,23 @@ saddles is found in Fig.~\ref{fig:franz-parisi}. As noted above, there is little qualitatively different from what was found in \cite{Ros_2019_Complexity} for the pure models. -\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. +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}. -\paragraph{Funding information} +\begin{figure} + \centering + \includegraphics{figs/franz_parisi.pdf} -JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the -INFN. + \caption{ + Comparison of the lowest-energy stationary points at overlap $q$ with a + reference minimum of $E_0=-1.71865<E_\mathrm{th}$ and + $\mu_0=6.1>\mu_\mathrm m$ (yellow, top), and the zero-temperature Franz--Parisi potential + with respect to the same reference minimum (blue, bottom). The two curves + coincide precisely at their minimum $q=0$ and at the local maximum $q\simeq0.5909$. + } \label{fig:franz-parisi} +\end{figure} \bibliographystyle{SciPost_bibstyle} \bibliography{2-point} |