From 32f23e83eb092c07a898a57354eecf510e5681d4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 19 Jun 2023 18:32:56 +0200
Subject: Some tweaks.

---
 2-point.tex | 212 ++++++++++++++++++++++++++++++++++++++----------------------
 1 file changed, 135 insertions(+), 77 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index ed60022..b596e3b 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 most of marginal states into a continuous `manifold.'
+  the arrangement most of marginal inherent states into a continuous `manifold.'
 \end{abstract}
 
 \tableofcontents
@@ -119,7 +119,12 @@ 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 with
-the energy level at which marginal states dominate the free energy
+the energy level at which thermodynamic states attached to marginal inherent states\footnote{
+  For this paper, which focuses on minima, we will take \emph{state} to mean
+  \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
 \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
@@ -147,13 +152,24 @@ When both points are saddles, we see the arrangement of barriers relative to
 each other, perhaps learning something about the geometry of the basins of
 attraction that they surround.
 
+More specifically, one \emph{reference} point is fixed with certain properties.
+Then, we compute the logarithm of the number of other points constrained to lie
+at a fixed overlap from the reference point. The fact of constraining the count
+to a fixed overlap produces constrained points with atypical properties. For
+instance, we will see that when the constrained overlap is made sufficiently
+large, the points so constrained tend to have an isolated eigenvalue pulled out
+of their spectrum, and the associated eigenvector is correlated with the
+direction of the reference point. Without the proximity constraint, such an
+isolated eigenvalue amounts to a large deviation from the typical spectrum of
+stationary points.
+
 In order to address the open problem of what attracts the long-time dynamics,
 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. Moreover, we show that the usual picture of a
-marginal `manifold' of states separated by subextensive barriers is only true
+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.
@@ -162,7 +178,7 @@ 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
 \S\ref{sec:complexity} we detail the calculation of the two-point complexity,
 and in \S\ref{sec:eigenvalue} and \S\ref{sec:franz-parisi} we do the same for
-the properties of the isolated eigenvalue and for the zero-temperature
+the properties of an isolated eigenvalue and for the zero-temperature
 Franz--Parisi potential.
 
 \section{Model}
@@ -211,23 +227,6 @@ The gradient and Hessian at a stationary point are then
 \end{align}
 where $\partial=\frac\partial{\partial\mathbf s}$ will always denote the derivative with respect to $\mathbf s$.
 
-When we count stationary points, we classify them by certain properties. One of
-these is the energy density $E=H/N$. We will also fix the \emph{stability}
-$\mu=\frac1N\operatorname{Tr}\operatorname{Hess}H$, also known as the radial
-reaction. In the mixed spherical models, all stationary points have a
-semicircle law for the eigenvalue spectrum of their Hessians, each with the
-same width $\mu_\mathrm m$, but whose center is shifted by different amounts. Fixing the
-stability $\mu$ fixes this shift, and therefore fixes the spectrum of the
-associated stationary point. When the stability is smaller than the width of
-the spectrum, or $\mu<\mu_\mathrm m$, there are an extensive number of negative
-eigenvalues, and the stationary point is a saddle with same large index whose
-value is set by the stability. When the stability is greater than the width of
-the spectrum, or $\mu>\mu_\mathrm m$, the semicircle distribution lies only
-over positive eigenvalues, and unless an isolated eigenvalue leaves the
-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
@@ -263,11 +262,37 @@ marginal minima.
   } \label{fig:spectra}
 \end{figure}
 
+When we count stationary points, we classify them by certain properties. One of
+these is the energy density $E=H/N$. We will also fix the \emph{stability}
+$\mu=\frac1N\operatorname{Tr}\operatorname{Hess}H$, also known as the radial
+reaction. In the mixed spherical models, all stationary points have a
+semicircle law for the eigenvalue spectrum of their Hessians, each with the
+same width $\mu_\mathrm m$, but whose center is shifted by different amounts. Fixing the
+stability $\mu$ fixes this shift, and therefore fixes the spectrum of the
+associated stationary point. When the stability is smaller than the width of
+the spectrum, or $\mu<\mu_\mathrm m$, there are an extensive number of negative
+eigenvalues, and the stationary point is a saddle with a large index whose
+value is set by the stability. When the stability is greater than the width of
+the spectrum, or $\mu>\mu_\mathrm m$, the semicircle distribution lies only
+over positive eigenvalues, and unless an isolated eigenvalue leaves the
+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. Fig.~\ref{fig:spectra} shows what different values of the
+stability imply about the spectrum at stationary points.
+
 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
 therefore only that energy has the possibility of trapping the long-time
-dynamics.
+dynamics. In generic mixed models this is not the case and at a given energy
+level $E$ there are many stabilities for which exponentially many stationary
+points are found. We define the threshold energy $E_\mathrm{th}$ as the energy
+at which most stationary points are marginal. Note that crucially this is
+\emph{not} the energy that has the most marginal stationary points: this energy
+level with the largest number of marginal points has even more saddles of
+extensive index! So $E_\mathrm{th}$ contains a \emph{minority} of the
+marginal points, even if those marginal points are the \emph{majority} of
+stationary points with energy $E_\mathrm{th}$.
 
 \begin{figure}
   \centering
@@ -300,6 +325,20 @@ Fig.~\ref{fig:complexities}.
 \section{Results}
 \label{sec:results}
 
+Our results are in the form of the two-point complexity, which is defined as
+the logarithm of the number of stationary points with energy $E_1$ and
+stability $\mu_1$ that lie at an overlap $q$ with a reference stationary point
+whose energy is $E_0$ and stability is $\mu_0$. When the complexity is
+positive, there are exponentially many stationary points with the given
+properties conditioned on the existence of the reference one. When it is zero,
+there are only order-one such points, and when it is negative there are
+exponentially few (effectively, none). In the examples below, the boundary of
+zero complexity between exponentially many and few points is often highlighted.
+Finally, as a result of the condition that the counted points lie with a given
+proximity to the reference point, their spectrum can be modified by the
+presence of an isolated eigenvalue, which can change the stability as in
+Fig.~\ref{fig:spectra}.
+
 \subsection{Barriers around deep states}
 
 \begin{figure}
@@ -378,62 +417,33 @@ energy.
     macroscopic gap in their overlap with nearby minima and saddles. The
     nearest stationary points are saddles with a single downward direction,
     and always have a higher energy density than the reference state.
+    Fig.~\ref{fig:marginal.prop.below} shows examples of the neighborhoods of
+    these marginal minima.
 
   \item \textbf{Energies above the threshold.} Marginal states have neighboring
     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.
+    density than the reference state. The nearest neighboring marginal states
+    have an overlap gap with the reference state.
+    Fig.~\ref{fig:marginal.prop.above} shows examples of the neighborhoods of
+    these marginal minima.
 
   \item \textbf{At the threshold energy.} Marginal states have neighboring
     stationary points at arbitrarily close distance, with a cubic pseudogap
-    in their complexity. The nearest ones are oriented saddle points with an
-    extensive number of downward directions and can be found at the same energy
-    density as the reference state.
+    in their complexity. The nearest ones include oriented saddle
+    points with an extensive number of downward directions, and oriented stable
+    an marginal minima. Though most of the nearest states are found at higher
+    energies, they can be found at the same energy density as the reference
+    state. Fig.~\ref{fig:marginal.prop.thres} shows examples of the
+    neighborhoods of these marginal states.
 \end{itemize}
 
-This leads us to some general conclusions. First, at all energy densities
-except at the threshold energy, \emph{marginal minima are separated by
-extensive energy barriers}. Therefore, the picture of a marginal
-\emph{manifold} of many (even all) marginal states lying arbitrarily close and
-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 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}
-  \includegraphics{figs/nearest_marginal_above.pdf}
-
-  \caption{
-    The neighborhood of marginal states at several energies above the threshold
-    energy. \textbf{Left:} The range of energies $E_1$ at which nearby states
-    are found. For any $E_0>E_\mathrm{th}$, there always exists a $q$
-    sufficiently close to one such that the nearby states have strictly greater
-    energy than the reference state. \textbf{Center:} The range of stabilities
-    $\mu_1$ at which nearby states are found. As below, there is always a
-    sufficiently large overlap beyond which all nearby states are saddle with
-    an extensive number of downward directions. \textbf{Right:} The range of
-    energies at which \emph{other} marginal states are found. Here, the more
-    darkly shaded regions denote where an isolated eigenvalue appears. Marginal
-    states above the threshold are always separated by a gap in their overlap.
-  } \label{fig:marginal.prop.above}
-\end{figure}
-
 \begin{figure}
   \includegraphics{figs/nearest_energies_below.pdf}
+  \hfill
   \includegraphics{figs/nearest_stabilities_below.pdf}
+  \hfill
   \includegraphics{figs/nearest_marginal_below.pdf}
 
   \caption{
@@ -449,12 +459,36 @@ will never change from one.
     are found. Here, the more darkly shaded regions denote where an isolated
     eigenvalue appears. Marginal states below the threshold are always
     separated by a gap in their overlap.
+  } \label{fig:marginal.prop.below}
+\end{figure}
+
+\begin{figure}
+  \includegraphics{figs/nearest_energies_above.pdf}
+  \hfill
+  \includegraphics{figs/nearest_stabilities_above.pdf}
+  \hfill
+  \includegraphics{figs/nearest_marginal_above.pdf}
+
+  \caption{
+    The neighborhood of marginal states at several energies above the threshold
+    energy. \textbf{Left:} The range of energies $E_1$ at which nearby states
+    are found. For any $E_0>E_\mathrm{th}$, there always exists a $q$
+    sufficiently close to one such that the nearby states have strictly greater
+    energy than the reference state. \textbf{Center:} The range of stabilities
+    $\mu_1$ at which nearby states are found. There is always a
+    sufficiently large overlap beyond which all nearby states are saddle with
+    an extensive number of downward directions. \textbf{Right:} The range of
+    energies at which \emph{other} marginal states are found. Here, the more
+    darkly shaded regions denote where an isolated eigenvalue appears. Marginal
+    states above the threshold are always separated by a gap in their overlap.
   } \label{fig:marginal.prop.above}
 \end{figure}
 
 \begin{figure}
   \includegraphics{figs/nearest_energies_thres.pdf}
+  \hfill
   \includegraphics{figs/nearest_stabilities_thres.pdf}
+  \hfill
   \includegraphics{figs/nearest_marginal_thres.pdf}
 
   \caption{
@@ -471,6 +505,25 @@ will never change from one.
   } \label{fig:marginal.prop.thres}
 \end{figure}
 
+This leads us to some general conclusions. First, at all energy densities
+except at the threshold energy, \emph{marginal minima are separated by
+extensive energy barriers}. Therefore, the picture of a marginal
+\emph{manifold} of many (even all) marginal states lying arbitrarily close and
+being connected by subextensive energy barriers can only describe the
+collection of marginal minima at the threshold energy. At energies 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 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' of nearby marginal minima within such a
+cluster cannot describe aging, since the overlap with the initial condition
+will never change from one.
+
 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
@@ -490,7 +543,7 @@ 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.
 
-\section{Complexity}
+\section{Calculation of the two-point complexity}
 \label{sec:complexity}
 
 We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
@@ -522,7 +575,7 @@ $E_1$ and stability $\mu_1$ that lie a fixed overlap $q$ from a reference
 stationary point of energy density $E_0$ and stability $\mu_0$. For a
 \emph{typical} number, we cannot average the total number $\mathcal N_H$, which
 is exponentially large in $N$ and therefore can be biased by atypical examples.
-Therefore, we will average the log of this number. The two-point complexity is
+Therefore, we will average the logarithm of this number. The two-point complexity is
 therefore defined by
 \begin{equation} \label{eq:complexity.definition}
   \Sigma_{12}
@@ -535,7 +588,7 @@ Both the denominator and the logarithm are treated using the replica trick, whic
   =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b,\varsigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb \sigma_1\cdot \mathbf s_a)\right)}
 \end{equation}
 Note that because of the structure of \eqref{eq:complexity.definition},
-$\pmb\sigma_1$ is special among the set of $\pmb\sigma$ replicas, since only it
+$\pmb\sigma_1$ is special among the set of $\pmb\sigma$ replicas, since it alone
 is constrained to lie a given overlap from the $\mathbf s$ replicas. This
 replica asymmetry will be important later.
 
@@ -554,7 +607,7 @@ which means that the matrix of partial derivatives belongs to the GOE class. Its
 \end{equation}
 with radius $\mu_\text m=\sqrt{4f''(1)}$. Since the Hessian differs from the
 matrix of partial derivatives by adding the constant diagonal matrix $\omega
-I$, it follows that the spectrum of the Hessian is a Winger semicircle shifted
+I$, it follows that the spectrum of the Hessian is a Wigner semicircle shifted
 by $\omega$, or $\rho(\lambda+\omega)$.
 
 The average over factors depending on the Hessian alone can be made separately
@@ -568,7 +621,7 @@ order in $N$ \cite{Ros_2019_Complexity}. At leading order, the various expectati
   \overline{\big|\det\operatorname{Hess}H(\mathbf s,\omega)\big|\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\omega)\big)}
   =e^{N\int d\lambda\,\rho(\lambda+\mu)\log|\lambda|}\delta(N\mu-N\omega)
 \end{equation}
-Therefore, all of the Lagrange multipliers are fixed identically to the stabilities $\mu$. We define the function
+Therefore, all of the Lagrange multipliers are fixed to the stabilities $\mu$. We define the function
 \begin{equation}
   \begin{aligned}
     \mathcal D(\mu)
@@ -589,9 +642,10 @@ and the full factor due to the Hessians is
 
 \subsection{The other factors}
 
-Having integrated over the Lagrange multipliers using the $\delta$ functions
-resulting from the average of the Hessians, the remaining part of the integrand
-has the form
+Having integrated over the Lagrange multipliers using the $\delta$-functions
+resulting from the average of the Hessians, any $\delta$-functions in the
+remaining integrand we Fourier transform into their integral representation
+over auxiliary fields. The resulting integrand has the form
 \begin{equation}
   e^{
     Nm\hat\beta_0E_0+Nn\hat\beta_1E_1
@@ -616,6 +670,10 @@ where we have introduced the linear operator
     i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}-\hat\beta_1
   \right)
 \end{equation}
+Here the $\hat\beta$s are the fields auxiliary to the energy constraints, the
+$\hat\mu$s are auxiliary to the spherical and overlap constraints, and the
+$\hat{\pmb\sigma}$s and $\hat{\mathbf s}$s are auxiliary to the constraint that
+the gradient be zero.
 We have written the $H$-dependant terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that
 \begin{equation}
   \overline{e^{\int d\mathbf t\,\mathcal O(\mathbf t)H(\mathbf t)}}
@@ -626,7 +684,7 @@ It remains only to apply the doubled operators to $f$ and then evaluate the simp
 
 \subsection{Hubbard--Stratonovich}
 
-Having expanded this expression, we are left with an argument in the exponential which is a function of scalar products between the fields $\mathbf s$, $\hat{\mathbf s}$, $\pmb\sigma$, and $\hat{\pmb\sigma}$. We will change integration coordinates from these fields to matrix fields given by the scalar products, defined as
+Having expanded this expression, we are left with an argument in the exponential which is a function of scalar products between the fields $\mathbf s$, $\hat{\mathbf s}$, $\pmb\sigma$, and $\hat{\pmb\sigma}$. We will change integration coordinates from these fields to matrix fields given by their scalar products, defined as
 \begin{equation} \label{eq:fields}
   \begin{aligned}
     C^{00}_{ab}=\frac1N\pmb\sigma_a\cdot\pmb\sigma_b &&
@@ -641,7 +699,7 @@ Having expanded this expression, we are left with an argument in the exponential
     D^{11}_{ab}=\frac1N\hat{\mathbf s}_a\cdot\hat{\mathbf s}_b
   \end{aligned}
 \end{equation}
-We insert into the integral the product of $\delta$ functions enforcing these
+We insert into the integral the product of $\delta$-functions enforcing these
 definitions, integrated over the new matrix fields, which is equivalent to
 multiplying by one. Once this is done, the many scalar products appearing
 throughout can be replaced by the matrix fields, and the original vector fields
@@ -665,7 +723,7 @@ D^{11})$, and $\mathcal Q_{01}=(\hat\mu_{01},C^{01},R^{01},R^{10},D^{01})$
 the resulting complexity is
 \begin{equation}
   \Sigma_{01}
-  =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int d\mathcal Q_{00}\,d\mathcal Q_{11}\,d\mathcal Q_{01}\,e^{Nm\mathcal S_0(\mathcal Q_{00})+Nn\mathcal S_1(\mathcal Q_{00},\mathcal Q_{11},\mathcal Q_{01})}
+  =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int d\mathcal Q_{00}\,d\mathcal Q_{11}\,d\mathcal Q_{01}\,e^{Nm\mathcal S_0(\mathcal Q_{00})+Nn\mathcal S_1(\mathcal Q_{11},\mathcal Q_{01}\mid\mathcal Q_{00})}
 \end{equation}
 where
 \begin{equation} \label{eq:one-point.action}
@@ -682,7 +740,7 @@ where
 is the action for the ordinary, one-point complexity, and remainder is given by
 \begin{equation} \label{eq:two-point.action}
   \begin{aligned}
-    &\mathcal S(\mathcal Q_{00},\mathcal Q_{11},\mathcal Q_{01})
+    &\mathcal S(\mathcal Q_{11},\mathcal Q_{01}\mid\mathcal Q_{00})
     =\hat\beta_1E_1-r^{11}_\mathrm d\mu_1-\frac12\hat\mu_1(1-c^{11}_\mathrm d)+\mathcal D(\mu_1) \\
     &\quad+\frac1n\sum_b^n\left\{-\frac12\hat\mu_{12}(q-C^{01}_{1b})+\sum_a^m\left[
         \hat\beta_0\hat\beta_1f(C^{01}_{ab})+(\hat\beta_0R^{01}_{ab}+\hat\beta_1R^{10}_{ab}-D^{01}_{ab})f'(C^{01}_{ab})+R^{01}_{ab}R^{10}_{ab}f''(C^{01}_{ab})
-- 
cgit v1.2.3-54-g00ecf

