1 files changed, 40 insertions, 18 deletions
diff --git a/2-point.tex b/2-point.tex
index 22cce5b..21d7dbf 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -110,11 +110,10 @@
 
 Many systems exhibit ``glassiness,'' characterized by rapid slowing of dynamics
 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
+spin glasses, certain inference and optimization problems, and more. Glassiness is qualitatively understood to arise from structure of
 an energy or cost landscape, whether due to the proliferation of metastable
-states, or to the raising of barriers which cause effective dynamic constraints \cite{Cavagna_2001_Fragile}.
-However, in most models there is no quantitative correspondence between these
+states, or to the raising of barriers which cause effective dynamic constraints \cite{Cavagna_2001_Fragile, Stillinger_2013_Glass, Kirkpatrick_2015_Colloquium}.
+However, in most models there is no known 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
@@ -124,7 +123,7 @@ the energy level at which thermodynamic states attached to marginal inherent sta
   \emph{minimum} or equivalently \emph{inherent state} and not a thermodynamic
   state. Any discussion of thermodynamic or equilibrium states will explicitly
   specify this.
-}dominate the free energy
+} 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
@@ -172,7 +171,7 @@ are not distinguished by this measure. Moreover, we show that the usual picture
 marginal `manifold' of inherent 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.
+dynamics this paper merely deepens the outstanding issues.
 
 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
@@ -316,11 +315,11 @@ In this study, we focus exclusively on the model studied in
   f_{3+4}(q)=\frac12\big(q^3+q^4\big)
 \end{equation}
 First, it has convex $f''(q)^{-1/2}$, so at least the ground state complexity
-must be replica symmetric, and second, properties of its long-time dynamics
-have been extensively studied. The annealed one-point complexity of these
-models was calculated in \cite{BenArous_2019_Geometry}, and for this model the
-annealed is expected to be correct. The complexity of this model is plotted in
-Fig.~\ref{fig:complexities}.
+must be replica symmetric, as in the pure spherical models. Second, properties
+of its long-time dynamics have been extensively studied. The annealed one-point
+complexity of these models was calculated in \cite{BenArous_2019_Geometry}, and
+for this model the annealed is expected to be correct. The complexity of this
+model is plotted in Fig.~\ref{fig:complexities}.
 
 \section{Results}
 \label{sec:results}
@@ -378,7 +377,7 @@ 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,
+saddles with an extensive index 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 $\mu_1$
@@ -407,8 +406,8 @@ lowest-energy states. This is seen in Fig.~\ref{fig:franz-parisi}.
 
 The set of marginal states is of special interest. First, it has more structure
 than in the pure models, with different types of marginal states being found at
-different energies. Second, these states attract the dynamics, and so are the
-inevitable end-point of equilibrium and algorithmic processes. We find,
+different energies. Second, these states attract the dynamics (as evidenced by power-law relaxations), and so are the
+inevitable end-point of equilibrium and algorithmic processes \cite{Folena_2023_On}. We find,
 surprisingly, that the properties of marginal states pivot around the threshold
 energy.
 
@@ -1376,7 +1375,7 @@ block constant matrices, things can be worked out from here. For instance, the
 second term in $M_{11}$ contributes nothing once the appropriate limits are
 taken, because each contribution is proportional to $n$.
 
-The contribution con be written as
+The contribution can be written as
 \begin{equation} \label{eq:inverse.quadratic.form}
     \begin{bmatrix}
       X_0\\i\hat X_0
@@ -1662,13 +1661,36 @@ 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 dynamical relaxation. 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 to 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.
+
 The methods developed in this paper are straightforwardly (if not easily)
-generalized to landscapes with replica symmetry broken complexities \cite{Kent-Dobias_2023_How}.
+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, and Chiara
-Cammarota for useful discussions related to this work.
+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}