From ae565b36c22ae41973dabca96be1311e80062119 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 4 Dec 2023 15:45:43 +0100
Subject: Big round of edits and reviewer responses.

---
 2-point.tex | 86 +++++++++++++++++++++++++++++++++++++++++++------------------
 1 file changed, 61 insertions(+), 25 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 18c5a6b..8f54a6a 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -383,9 +383,11 @@ Fig.~\ref{fig:spectra}.
     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 line on the left plot depicts the
-    trajectory of the solid line on the right plot, and the dot-dashed line on the right plot likewise depicts the solid line on the left 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.
+    trajectory of the solid line on the right plot, and the dot-dashed line on
+    the right plot depicts the trajectory of the solid line on the left 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, which makes them an atypical population of saddles.
   } \label{fig:min.neighborhood}
 \end{figure}
 
@@ -397,7 +399,7 @@ 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}.
 
-The largest different is the decoupling of nearby
+The largest difference is the decoupling of nearby
 stable points from nearby low-energy points: in the pure $p$-spin model, the
 left and right panels of Fig.~\ref{fig:min.neighborhood} would be identical up
 to a constant factor $-p$. Instead, for mixed models they differ substantially,
@@ -406,11 +408,11 @@ would correspond exactly with the solid lines. One significant consequence of
 this difference is the diminished significance of the threshold energy
 $E_\text{th}$: in the left panel, marginal minima of the threshold energy are
 the most common among unconstrained points, but marginal minima of lower energy
-are more common in the near vicinity of the example reference minimum, whose energy is lower than the threshold energy.
+are more common in the near vicinity of the example reference minimum, whose energy is lower than the threshold energy. In the pure models, all marginal minima are at the threshold energy.
 
 The nearest neighbor points are always oriented saddles, sometimes
 saddles with an extensive index and sometimes index-one saddles
-(Fig.~\ref{fig:spectra}(d, f)). Like in the pure models, the minimum energy and
+(Fig.~\ref{fig:spectra}(d, f)). This is a result of the persistent presence of a negative isolated eigenvalue in the spectrum of the nearest neighbors, e.g., as in the shaded regions of Fig.~\ref{fig:min.neighborhood}. Like in the pure models, the minimum energy and
 maximum stability of nearby points are not monotonic: there is a range of
 overlap where the minimum energy of neighbors decreases with proximity. The
 emergence of oriented index-one saddles along the line of lowest-energy states
@@ -552,7 +554,7 @@ extensive energy barriers}. Therefore, the picture of a marginal
 being connected by subextensive energy barriers can only describe the
 collection of marginal minima at the threshold energy, which is an atypical population of marginal minima. At energies both below and above the threshold energy,
 typical marginal minima are isolated from each other.\footnote{
-We must put a small caveat here: in \emph{any} situation, this calculation
+We must put a small caveat here: for any combination of energy and stability of the reference point, 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
@@ -772,8 +774,7 @@ To examine better the population of marginal points, it is necessary to look at
 the next term in the series of the complexity with $\Delta q$, since the linear
 coefficient becomes zero at the marginal line. This tells us something
 intuitive: stable minima have an effective repulsion between points, and one
-always finds a sufficiently small $\Delta q$ such that no stationary points are
-point any nearer. For the marginal minima, it is not clear that the same should be true.
+always finds a sufficiently small $\Delta q$ such that no stationary points are found any nearer. For the marginal minima, it is not clear that the same should be true.
 
 For marginal points with $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion for the dominant population of near neighbors is
 \begin{equation}
@@ -1191,7 +1192,7 @@ fields the Hessian is independent of these \cite{Bray_2007_Statistics}. In
 principle the fact that we have conditioned the Hessian to belong to stationary
 points of certain energy, stability, and proximity to another stationary point
 will modify its statistics, but these changes will only appear at subleading
-order in $N$ \cite{Ros_2019_Complexity}. This is because the conditioning amounts to a rank-one perturbation to the Hessian matrix. At leading order, the expectations related to different replicas factorize, each yielding
+order in $N$ \cite{Ros_2019_Complexity}. This is because the conditioning amounts to a rank-one perturbation to the Hessian matrix, which does not affect the bulk of its spectrum. At leading order, the expectations related to different replicas factorize, each yielding
 \begin{equation}
   \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)
@@ -1259,7 +1260,12 @@ We have written the $H$-dependent terms in this strange form for the ease of tak
   =e^{\frac12\int d\mathbf t\,d\mathbf t'\,\mathcal O(\mathbf t)\mathcal O(\mathbf t')\overline{H(\mathbf t)H(\mathbf t')}}
   =e^{N\frac12\int d\mathbf t\,d\mathbf t'\,\mathcal O(\mathbf t)\mathcal O(\mathbf t')f\big(\frac{\mathbf t\cdot\mathbf t'}N\big)}
 \end{equation}
-It remains only to apply the doubled operators to $f$ and then evaluate the simple integrals over the $\delta$ measures. We do not include these details, which were carried out with computer algebra software.
+It remains only to apply the doubled operators to $f$ and then evaluate the
+simple integrals over the $\delta$ measures. We do not include these details,
+which were carried out with computer algebra software. The result of this
+calculation is found in the effective action \eqref{eq:intermed.complexity},
+where it contributes all terms besides the functions $\mathcal D$ contributed by the Hessian terms in the previous section and the
+logarithms contributed by the Hubbard--Stratonovich transformation of the next section.
 
 \subsection{Hubbard--Stratonovich}
 \label{subsec:hubbard.strat}
@@ -1312,7 +1318,16 @@ The integral over the vector fields $\mathbf a$ is Gaussian and can be evaluated
 Finally, the integral over $\hat Q$ can be evaluated using the saddle point
 method, giving $\hat Q=Q^{-1}$. Therefore, the term contributed to the effective
 action in the matrix fields as a result of the transformation is
-$N(\frac12+\frac12\log\det Q)$.
+\begin{equation}
+  \frac12\log\det Q
+  =
+  \frac12\log\det\begin{bmatrix}
+    C^{00}&iR^{00}&C^{01}&iR^{01}\\
+    iR^{00}&D^{00}&iR^{10}&D^{01}\\
+    C^{01}&iR^{10}&C^{11}&iR^{11}\\
+    iR^{01}&D^{01}&iR^{11}&D^{11}
+  \end{bmatrix}
+\end{equation}
 
 \subsection{Replica ansatz and saddle point}
 \label{subsec:saddle}
@@ -1340,7 +1355,7 @@ Defining the `block' fields $\mathcal Q_{00}=(\hat\beta_0, \hat\mu_0, C^{00},
 R^{00}, D^{00})$, $\mathcal Q_{11}=(\hat\beta_1, \hat\mu_1, C^{11}, R^{11},
 D^{11})$, and $\mathcal Q_{01}=(\hat\mu_{01},C^{01},R^{01},R^{10},D^{01})$
 the resulting complexity is
-\begin{equation}
+\begin{equation} \label{eq:intermed.complexity}
   \Sigma_{12}
   =\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}
@@ -1352,7 +1367,7 @@ where
     &\quad+\frac1m\bigg\{
       \frac12\sum_{ab}^m\left[
         \hat\beta_1^2f(C^{00}_{ab})+(2\hat\beta_1R^{00}_{ab}-D^{00}_{ab})f'(C^{00}_{ab})+(R_{ab}^{00})^2f''(C_{ab}^{00})
-  \right]+\frac12\log\det\begin{bmatrix}C^{00}&R^{00}\\R^{00}&D^{00}\end{bmatrix}
+  \right]+\frac12\log\det\begin{bmatrix}C^{00}&iR^{00}\\iR^{00}&D^{00}\end{bmatrix}
 \bigg\}
 \end{aligned}
 \end{equation}
@@ -1896,22 +1911,48 @@ between the eigenvector $\mathbf x_\text{min}$ associated with the minimum eigen
 direction connecting the two stationary points $\mathbf x_{0\leftarrow1}$.
 The overlap between these vectors is directly related to the value of the order parameter $x_0=\frac1N\pmb\sigma_1\cdot\mathbf x_a$. This tangent vector is $\mathbf x_{0\leftarrow
 1}=\frac1{1-q}\big(\pmb\sigma_1-q\mathbf s_a\big)$, which is normalized and
-lies strictly in the tangent plane of $\mathbf s_a$. Then
+lies strictly in the tangent plane of $\mathbf s_a$. Then the overlap between the two vectors is
 \begin{equation}
   q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N
   =\frac{x_0}{1-q}
 \end{equation}
-where $\mathrm x_\text{min}\cdot\mathrm s_a=0$ because of the restriction of
-the $\mathrm x$ vectors to the tangent plane at $\mathrm s_a$.
+where $\mathbf x_\text{min}\cdot\mathbf s_a=0$ because of the restriction of
+the $\mathbf x$ vectors to the tangent plane at $\mathbf s_a$.
 
 
 \section{Comparison with the Franz--Parisi potential}
 \label{sec:franz-parisi}
 
-Here, we compute the Franz--Parisi potential for this model at zero
+The comparison between the Franz--Parisi potential at zero temperature and the
+minimum-energy limit of the two-point complexity is of interest to some
+specialists because the two computations qualitatively describe the same thing.
+However, it was previously found that the two computations produce different
+results in the pure spherical models, to the surprise of those researchers
+\cite{Ros_2019_Complexity}. Understanding this difference is subtle. The
+zero-temperature Franz--Parisi potential underestimates the energy where nearby
+minima are found, because it includes any configuration that is a minimum on
+the subspace created by constraining the overlap. Many of these configurations
+will not have zero gradient perpendicular to the overlap constraint manifold,
+and therefore are not proper minima of the energy.
+
+A strange feature of the comparison for the pure spherical models was that the
+two-point complexity and the Franz--Parisi potential coincided at their local
+maximum in $q$. It is not clear why this coincidence occurs, but it is good
+news for those who use the Franz--Parisi potential to estimate the height of
+the free energy barrier between states. Though it everywhere else
+underestimates the energy of nearby states, it correctly gives the value of
+this highest barrier.
+
+Here, we compute the Franz--Parisi potential for the mixed spherical models at zero
 temperature, with respect to a reference configuration fixed to be a stationary
-point of energy $E_0$ and stability $\mu_0$ as before \cite{Franz_1995_Recipes,
-Franz_1998_EffectivePotential}. The potential is defined as the average free
+point of energy $E_0$ and stability $\mu_0$ \cite{Franz_1995_Recipes,
+Franz_1998_EffectivePotential}. Comparing with the lower energy boundary of the
+2-point complexity, we find that the story in the mixed models is the same as
+that in the pure models: the Franz--Parisi potential underestimates the lowest
+energy of nearby minima almost everywhere except at its peak, where the two
+measures coincide.
+
+The potential is defined as the average free
 energy of a system constrained to lie with a fixed overlap $q$ with a reference
 configuration (here a stationary point with fixed energy and stability), and
 given by
@@ -2001,11 +2042,6 @@ 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.
 
-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}
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