From cca6ae689762dd447a464be37a2b5229248235e1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 28 Nov 2023 17:00:33 +0100
Subject: Big text reshuffle as more is put in the appendix.

---
 2-point.tex | 1284 ++++++++++++++++++++++++++++++-----------------------------
 1 file changed, 643 insertions(+), 641 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 9be13e9..ce2d1bc 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -612,718 +612,720 @@ $\pmb\sigma_1$ 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.
 
-\subsection{The Hessian factors}
-
-The double partial derivatives of the energy are Gaussian with the variance
-\begin{equation}
-  \overline{(\partial_i\partial_jH(\mathbf s))^2}=\frac1Nf''(1)
-\end{equation}
-which means that the matrix of partial derivatives belongs to the GOE class. Its spectrum is given by the Wigner semicircle
+The resulting expression for the complexity, which must
+still be extremized over the parameters $\hat\beta_1$, $r^{01}$,
+$r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is
 \begin{equation}
-  \rho(\lambda)=\begin{cases}
-    \frac2{\pi}\sqrt{1-\big(\frac{\lambda}{\mu_\text m}\big)^2} & \lambda^2\leq\mu_\text m^2 \\
-    0 & \text{otherwise}
-  \end{cases}
+  \begin{aligned}
+    &\Sigma_{12}(E_0,\mu_0,E_1,\mu_1,q)
+    =\mathop{\mathrm{extremum}}_{\hat\beta_1,r^{11}_\mathrm d,r^{11}_0,r^{01},q^{11}_0}\Bigg\{
+    \mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1
+    +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)\\
+    &\qquad+\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q)
+    +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)\\
+    &+\frac12\Bigg[
+      \hat\beta_1^2\big(f(1)-f(q^{11}_{0})\big)
+      +(r^{11}_\mathrm d)^2f''(1)+2r^{01}r^{10}f''(q)-(r^{11}_0)^2f''(q^{11}_0)
+      +\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1)
+      \\
+    &\qquad+\frac{1-q^2}{1-q^{11}_0}+\left(
+      (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}
+      \left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right)
+    \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \\
+    &\qquad
+    -\frac1{f'(1)}\frac{f'(1)^2-f'(q)^2}{f'(1)-f'(q^{11}_0)}
+    +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)
+    +\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right)
+  \Bigg]\Bigg\}
+  \end{aligned}
 \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 Wigner semicircle shifted
-by $\omega$, or $\rho(\lambda+\omega)$.
+It is possible to further extremize this expression over all the other
+variables but $q_0^{11}$, for which the saddle point conditions have a unique
+solution. However, the resulting expression is quite complicated and provides
+no insight. In practice, the complexity can be calculated in two ways. First,
+the extremal problem can be done numerically, initializing from $q=0$ where the
+problem reduces to that of the single-point complexity of points with energy
+$E_1$ and stability $\mu_1$, and then taking small steps in $q$ or other
+parameters to trace out the solution. This is how the data in all the plots of
+this paper was produced. Second, the complexity can be calculated in the near
+neighborhood of a reference point by expanding in small $1-q$. This is what we
+describe in the next subsection.
 
-The average over factors depending on the Hessian alone can be made separately
-from those depending on the gradient or energy, since for random Gaussian
-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}. At leading order, the various expectations factorize, each yielding
+If there is no overlap gap between the reference point and its nearest
+neighbors, their complexity can be calculated by an expansion in $1-q$. First,
+we'll use this method to describe the most common type of stationary point in
+the close vicinity of a reference point. The most common neighbors of a
+reference point are given by further maximizing the two-point complexity over
+the energy $E_1$ and stability $\mu_1$ of the nearby points. This gives the
+conditions
+\begin{align}
+  \hat\beta_1=0 &&
+  \mu_1=2r^{11}_\mathrm df''(1)
+\end{align}
+where the second is only true for $\mu_1^2\leq\mu_\mathrm m^2$, i.e., when the
+nearby points are saddle points or marginal minima. When these conditions are
+inserted into the complexity, an expansion is made in small $1-q$, and the
+saddle point in the remaining parameters is taken, the result is
 \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)
+  \Sigma_{12}
+  =\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)\big(f''(1)-f'(1)\big)}{f'''(1)f'(1)}}-1\right)(1-q)
+  +O\big((1-q)^2\big)
 \end{equation}
-Therefore, all of the Lagrange multipliers are fixed to the stabilities $\mu$. We define the function
+independent of $E_0$. To describe the properties of these most common
+neighbors, it is convenient to first make a definition. The population of
+stationary points that are most common at each energy (the blue line in
+Fig.~\ref{fig:complexities}) have the relation
 \begin{equation}
-  \begin{aligned}
-    \mathcal D(\mu)
-    &=\int d\lambda\,\rho(\lambda+\mu)\log|\lambda| \\
-    &=\begin{cases}
-      \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
-       & \mu^2\leq\mu_\text m^2 \\
-      \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
-      -\left|\frac{\mu}{\mu_\text m}\right|\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}
-      -\log\left(\left|\frac{\mu}{\mu_\text m}\right|-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu^2>\mu_\text m^2
-    \end{cases}
-  \end{aligned}
+  E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0
 \end{equation}
-and the full factor due to the Hessians is
+between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this
+definition, the energy and stability of the most common neighbors at small
+$\Delta q$ are
+\begin{align} \label{eq:expansion.E.1}
+  E_1&=E_0+\frac12\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\
+    \label{eq:expansion.mu.1}
+  \mu_1&=\mu_0-\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big)
+\end{align}
+The most common neighboring saddles to a reference saddle are much nearer to
+the reference in energy ($\Delta q^2$) than in stability ($\Delta q$). In fact,
+this scaling also holds for the entire range of neighbors to a reference
+saddle, with the limits in energy scaling like $\Delta q^2$ and those of
+stability scaling like $\Delta q$.
+
+Because both expressions are proportional to $E_0-E_\mathrm{dom}(\mu_0)$,
+whether the energy and stability of nearby points increases or decreases from
+that of the reference point depends only on whether the energy of the reference
+point is above or below that of the most common population at the same
+stability. In particular, since $E_\mathrm{dom}(\mu_\mathrm m)=E_\mathrm{th}$,
+the threshold energy is also the pivot around which the points asymptotically
+nearby marginal minima change their properties.
+
+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 $\Delta q$ that no stationary points are
+point any nearer. For the marginal minima, it is not clear that the same should be true.
+
+When $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion is
 \begin{equation}
-  e^{Nm\mathcal D(\mu_0)+Nn\mathcal D(\mu_1)}\left[\prod_a^m\delta(N\mu_0-N\varsigma_a)\right]\left[\prod_a^n\delta(N\mu_1-N\omega_a)\right]
+  \Sigma_{12}
+  =\frac12\frac{f'''(1)v_f}{f''(1)^{3/2}u_f}
+  \left(\sqrt{\frac{2\big[f'(1)(f'''(1)-f''(1))+f''(1)^2\big]}{f'(1)f'''(1)}}-1\right)\big(E_0-E_\textrm{th}\big)(1-q)^2+O\big((1-q)^3\big)
 \end{equation}
+Note that this expression is only true for $\mu=\mu_\mathrm m$. Therefore,
+among marginal minima, when $E_0$ is greater than the threshold one finds
+neighbors at arbitrarily close distance. When $E_0$ is less than the threshold,
+the complexity of nearby points is negative, and there is a desert where none
+are found.
 
-\subsection{The other factors}
+\begin{figure}
+  \centering
+  \includegraphics{figs/expansion_energy.pdf}
+  \hspace{1em}
+  \includegraphics{figs/expansion_stability.pdf}
 
-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
+  \caption{
+    Demonstration of the convergence of the $(1-q)$-expansion for marginal
+    reference minima. Solid lines and shaded region show are the same as in
+    Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$.
+    The dotted lines show the expansion of most common neighbors, while the
+    dashed lines in both plots show the expansion for the minimum and maximum
+    energies and stabilities found at given $q$.
+  } \label{fig:expansion}
+\end{figure}
+
+The properties of the nearby states above the threshold can be
+further quantified. The most common points are still given by
+\eqref{eq:expansion.E.1} and \eqref{eq:expansion.mu.1}, but the range of
+available points can also be computed, and one finds that the stability lies in
+the range
+$\mu_1=\mu_\mathrm m+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$
+where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1}
+and
 \begin{equation}
-  e^{
-    Nm\hat\beta_0E_0+Nn\hat\beta_1E_1
-    -\sum_a^m\left[(\pmb\sigma_a\cdot\hat{\pmb\sigma}_a)\mu_0
-      -\frac12\hat\mu_0(N-\pmb\sigma_a\cdot\pmb\sigma_a)
-    \right]
-      -\sum_a^n\left[(\mathbf s_a\cdot\hat{\mathbf s}_a)\mu_1
-      -\frac12\hat\mu_1(N-\mathbf s_a\cdot\mathbf s_a)
-      -\frac12\hat\mu_{12}(Nq-\pmb\sigma_1\cdot\mathbf s_a)
-    \right]
-    +\int d\mathbf t\,\mathcal O(\mathbf t)H(\mathbf t)
+  \delta\mu_2=\frac{v_f}{f'(1)f''(1)^{3/4}}\sqrt{
+    \frac{E_0-E_\mathrm{th}}2\frac{2f''(1)\big(f''(1)-f'(1)\big)+f'(1)f'''(1)}{u_f}
   }
 \end{equation}
-where we have introduced the linear operator
+Similarly, one finds that the energy lies in the range $E_1=E_0+\delta
+E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given
+by the coefficient in \eqref{eq:expansion.E.1} and
 \begin{equation}
-  \mathcal O(\mathbf t)
-  =\sum_a^m\delta(\mathbf t-\pmb\sigma_a)\left(
-    i\hat{\pmb\sigma}_a\cdot\partial_{\mathbf t}-\hat\beta_0
-  \right)
-  +
-  \sum_a^n\delta(\mathbf t-\mathbf s_a)\left(
-    i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}-\hat\beta_1
-  \right)
+  \begin{aligned}
+    \delta E_2
+    &=\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\bigg(
+      \frac{
+        v_f
+        }{3u_f}
+        \big[
+          f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))-2f''(1)^2
+        \big]
+        \\
+    &\hspace{12pc}\times
+        \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)-6f''(1)^2
+        \big]
+    \bigg)^\frac12
+  \end{aligned}
 \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 $H$-dependent terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that
+and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic equation
 \begin{equation}
-  \overline{e^{\int d\mathbf t\,\mathcal O(\mathbf t)H(\mathbf t)}}
-  =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)}
+  0=((16-(6-\delta q_0)\delta q_0)\delta q_0-12)f'(1)f'''(1)-2\delta q_0(f''(1)-f'(1))f''(1)
 \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 are standard.
-
-\subsection{Hubbard--Stratonovich}
+These predictions from the small $1-q$ expansion are compared with numeric
+saddle points for the complexity of marginal minima in
+Fig.~\ref{fig:expansion}, and the results agree well at small $1-q$.
 
-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 &&
-    R^{00}_{ab}=-i\frac1N{\pmb\sigma}_a\cdot\hat{\pmb\sigma}_b &&
-    D^{00}_{ab}=\frac1N\hat{\pmb\sigma}_a\cdot\hat{\pmb\sigma}_b \\
-    C^{01}_{ab}=\frac1N\pmb\sigma_a\cdot\mathbf s_b &&
-    R^{01}_{ab}=-i\frac1N{\pmb\sigma}_a\cdot\hat{\mathbf s}_b &&
-    R^{10}_{ab}=-i\frac1N\hat{\pmb\sigma}_a\cdot{\mathbf s}_b &&
-    D^{01}_{ab}=\frac1N\hat{\pmb\sigma}_a\cdot\hat{\mathbf s}_b \\
-    C^{11}_{ab}=\frac1N\mathbf s_a\cdot\mathbf s_b &&
-    R^{11}_{ab}=-i\frac1N{\mathbf s}_a\cdot\hat{\mathbf s}_b &&
-    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
-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
-can be integrated over. Conjugate matrix field integrals created when the
-$\delta$-functions are promoted to exponentials can be evaluated by saddle
-point in the standard way, yielding an effective action depending on the above
-matrix fields alone.
 
-\subsection{Saddle point}
+\section{Isolated eigenvalue}
+\label{sec:eigenvalue}
 
-We will always assume that the square matrices $C^{00}$, $R^{00}$, $D^{00}$,
-$C^{11}$, $R^{11}$, and $D^{11}$ are hierarchical matrices, with each set of
-three sharing the same hierarchical structure. In particular, we immediately
-define $c_\mathrm d^{00}$, $r_\mathrm d^{00}$, $d_\mathrm d^{00}$, $c_\mathrm d^{11}$, $r_\mathrm d^{11}$, and
-$d_\mathrm d^{11}$ as the value of the diagonal elements of these matrices,
-respectively. Note that $c_\mathrm d^{00}=c_\mathrm d^{11}=1$ due to the spherical constraint.
+The two-point complexity depends on the spectrum at both stationary points
+through the determinant of their Hessians, but only on the bulk of the
+distribution. As we saw, this bulk is unaffected by the conditions of energy
+and proximity. However, these conditions give rise to small-rank perturbations
+to the Hessian, which can lead a subextensive number of eigenvalues leaving the
+bulk. We study the possibility of \emph{one} stray eigenvalue.
 
-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
+We use a technique recently developed to find the smallest eigenvalue of a
+random matrix \cite{Ikeda_2023_Bose-Einstein-like}. One defines a quadratic
+statistical mechanics model with configurations defined on the sphere, whose
+interaction tensor is given by the matrix of interest. By construction, the
+ground state is located in the direction of the eigenvector associated with the
+smallest eigenvalue, and the ground state energy is proportional to that
+eigenvalue.
+
+\begin{figure}
+  \centering
+  \begin{tikzpicture}
+    \def\R{4 } % sphere radius
+    \def\Rt{2 } % tangent plane radius
+    \def\angEl{15} % elevation angle
+    \def\angsa{-160} % azimuth of s_1
+    \def\angq{40} % elevation of constraint circle
+    \filldraw[ball color=white] (0,0) circle (\R);
+  %  \filldraw[fill=white] (0,0) circle (\R);
+
+    \foreach \t in {0,\angq} { \DrawLatitudeCircle[\R]{\t} }
+    %\foreach \t in {\angsa} { \DrawLongitudeCircle[\R]{\t} }
+
+    \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
+    \coordinate (O) at (0,0);
+    \node[circle,draw,black,scale=0.3] at (0,0) {};
+    \coordinate (N) at (0,\H);
+    \draw node[right=10,below] at (0,\H){$\pmb\sigma_1$};
+    \draw[thick, ->](O)--(N);
+
+    \NewLatitudePlane[planeP]{\R}{\angEl}{\angq};
+    \path[planeP] (\angsa:\R) coordinate (P);
+    \path[planeP] (0:1.5*\R) coordinate (Q);
+    \path[planeP] (0:\R) coordinate (Q2);
+    \draw[left] node at (P){$\mathbf s_1$};
+
+    \NewLatitudePlane[equator]{\R}{\angEl}{00};
+    \path[equator] (-30:\R) coordinate (Pprime);
+    \path[equator] (0:{1.5*cos(\angq)*\R}) coordinate (Qe);
+    \path[equator] (0:\R) coordinate (Qe2);
+    \draw node[right=5,below] at (Pprime){$\pmb\sigma_c$};
+
+    \NewLatitudePlane[sbplane]{\R}{\angEl}{\angq};
+    \path[sbplane] (20:\R) coordinate (sb);
+    \draw node[right=3,above=1] at (sb){$\mathbf s_b$};
+
+    \TangentPlane[tplane]{\R}{\angEl}{\angq}{\angsa};
+    \draw[tplane,fill=gray,fill opacity=0.3] circle (\Rt);
+    \draw[tplane,->,thick] (0,0) -> ({\Rt*cos(160)},{\Rt*sin(160)}) node[above=1.5,right] {$\mathbf x_a$};
+    \draw[tplane,->,thick] (0,0) -> ({\Rt*cos(250)},{\Rt*sin(250)}) node[above=1.5,left=0.1] {$\mathbf x_b$};
+
+    \draw[thick, ->] (O)->(P);
+    \draw[thick, ->] (O)->(Pprime);
+    \draw[thick, ->] (O)->(sb);
+
+    \draw[dotted] (Qe) -- (Qe2);
+    \draw[dotted] (Q2) -- (Q);
+    \draw[decorate, decoration = {brace,raise=3}] (Q) -- (Qe) node[pos=0.5,right=7]{$q$};
+  \end{tikzpicture}
+  \caption{
+    A sketch of the vectors involved in the calculation of the isolated
+    eigenvalue. All replicas $\mathbf x$, which correspond with candidate
+    eigenvectors of the Hessian evaluated at $\mathbf s_1$, sit in an $N-2$
+    sphere corresponding with the tangent plane (not to scale) of the first
+    $\mathbf s$ replica. All of the $\mathbf s$ replicas lie on the sphere,
+    constrained to be at fixed overlap $q$ with the first of the $\pmb\sigma$
+    replicas, the reference configuration. All of the $\pmb\sigma$ replicas lie
+    on the sphere.
+  } \label{fig:sphere}
+\end{figure}
+
+Our matrix of interest is the Hessian evaluated at a stationary point of the mixed spherical
+model, conditioned on the relative position, energies, and stabilities
+discussed above. We must restrict the artificial spherical model to lie in the
+tangent plane of the `real' spherical configuration space at the point of
+interest, to avoid our eigenvector pointing in a direction that violates the
+spherical constraint. A sketch of the setup is shown in Fig.~\ref{fig:sphere}. The free energy of this model given a point $\mathbf s$
+and a specific realization of the disordered Hamiltonian 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_{11},\mathcal Q_{01}\mid\mathcal Q_{00})}
+  \begin{aligned}
+    \beta F_H(\beta\mid\mathbf s,\omega)
+    &=-\frac1N\log\left(\int d\mathbf x\,\delta(\mathbf x\cdot\mathbf s)\delta(\|\mathbf x\|^2-N)\exp\left\{
+        -\beta\frac12\mathbf x^T\operatorname{Hess}H(\mathbf s,\omega)\mathbf x
+    \right\}\right) \\
+    &=-\lim_{\ell\to0}\frac1N\frac\partial{\partial\ell}\int\left[\prod_{\alpha=1}^\ell d\mathbf x_\alpha\,\delta(\mathbf x_\alpha^T\mathbf s)\delta(N-\mathbf x_\alpha^T\mathbf x_\alpha)\exp\left\{
+      -\beta\frac12\mathbf x^T_\alpha\big(\partial\partial H(\mathbf s)+\omega I\big)\mathbf x_\alpha
+    \right\}\right]
+  \end{aligned}
 \end{equation}
-where
-\begin{equation} \label{eq:one-point.action}
+where the first $\delta$-function keeps the configurations in the tangent
+plane, and the second enforces the spherical constraint. We have anticipated
+treating the logarithm with replicas. We are interested in points $\mathbf s$
+that have certain properties: they are stationary points of $H$ with given
+energy density and stability, and fixed overlap from a reference configuration
+$\pmb\sigma$. We therefore average the free energy above over such points,
+giving
+\begin{equation}
   \begin{aligned}
-    &\mathcal S_0(\mathcal Q_{00})
-    =\hat\beta_0E_0-r^{00}_\mathrm d\mu_0-\frac12\hat\mu_0(1-c^{00}_\mathrm d)+\mathcal D(\mu_0)\\
-    &\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}
-\bigg\}
-\end{aligned}
+    F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma)
+    &=\int\frac{d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s)}{\int d\nu_H(\mathbf s',\omega'\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s')}F_H(\beta\mid\mathbf s,\omega) \\
+    &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F_H(\beta\mid\mathbf s_1,\omega_1)
+  \end{aligned}
 \end{equation}
-is the action for the ordinary, one-point complexity, and remainder is given by
-\begin{equation} \label{eq:two-point.action}
+again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability. Averaging over these conditions gives
+\begin{equation}
   \begin{aligned}
-    &\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})
-    \right]\right\}
-    \\
-    &\quad+\frac1n\bigg\{
-      \frac12\sum_{ab}^n\left[
-        \hat\beta_1^2f(C^{11}_{ab})+(2\hat\beta_1R^{11}_{ab}-D^{11}_{ab})f'(C^{11}_{ab})+(R^{11}_{ab})^2f''(C^{11}_{ab})
-      \right]\\
-    &\quad+\frac12\log\det\left(
-      \begin{bmatrix}
-        C^{11}&iR^{11}\\iR^{11}&D^{11}
-      \end{bmatrix}-
-      \begin{bmatrix}
-        C^{01}&iR^{01}\\iR^{10}&D^{01}
-      \end{bmatrix}^T
-      \begin{bmatrix}
-        C^{00}&iR^{00}\\iR^{00}&D^{00}
-      \end{bmatrix}^{-1}
-      \begin{bmatrix}
-        C^{01}&iR^{01}\\iR^{10}&D^{01}
-      \end{bmatrix}
-    \right)
-  \bigg\}
+    F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)
+    &=\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma) \\
+    &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu_H(\pmb\sigma_a,\varsigma_a\mid E_0,\mu_0)\right]\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma_1)
   \end{aligned}
 \end{equation}
-Because of the structure of this problem in the twin limits of $m$ and $n$ to
-zero, the parameters $\mathcal Q_{00}$ can be evaluated at a saddle point of
-$\mathcal S_0$ alone. This means that these parameters will take the same value
-they take when the ordinary, 1-point complexity is calculated. For a replica
-symmetric complexity of the reference point, this results in
-\begin{align}
-  \hat\beta_0
-  &=-\frac{\mu_0f'(1)+E_0\big(f'(1)+f''(1)\big)}{u_f}\\
-  r_\mathrm d^{00}
-  &=\frac{\mu_0f(1)+E_0f'(1)}{u_f} \\
-  d_\mathrm d^{00}
-  &=\frac1{f'(1)}
-  -\left(
-    \frac{\mu_0f(1)+E_0f'(1)}{u_f}
-  \right)^2
-\end{align}
-where we define for brevity (here and elsewhere) the constants
-\begin{align}
-  u_f=f(1)\big(f'(1)+f''(1)\big)-f'(1)^2
-  &&
-  v_f=f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2
-\end{align}
-Note that because the coefficients of $f$ must be nonnegative for $f$ to
-be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. Note also
-that $u_f=v_f=0$ if $f$ is a homogeneous polynomial as in the pure models.
-These expressions are invalid for the pure models because $\mu_0$ and $E_0$
-cannot be fixed independently; we would have done the equivalent of inserting
-two identical $\delta$-functions. For the pure models, the terms $\hat\beta_0$ and
-$\hat\beta_1$ must be set to zero in our prior formulae (as if the energy was
-not constrained) and then the saddle point taken.
-
+This formidable expression is now ready to be averaged over the disordered Hamiltonians $H$. Once averaged,
+the minimum eigenvalue of the conditioned Hessian is then given by twice the ground state energy, or
+\begin{equation}
+  \lambda_\text{min}=2\lim_{\beta\to\infty}\overline{F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)}
+\end{equation}
+For this calculation, there are three different sets of replicated variables.
+Note that, as for the computation of the complexity, the $\pmb\sigma_1$ and
+$\mathbf s_1$ replicas are \emph{special}. The first again is the only of the
+$\pmb\sigma$ 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 general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$,
-and $D^{01}$ to have constant \emph{rows} of length $n$, with blocks of rows
-corresponding to the \textsc{rsb} structure of the single-point complexity. For
-the scope of this paper, where we restrict ourselves to replica symmetric
-complexities, they have the following form at the saddle point:
-\begin{align} \label{eq:01.ansatz}
-  C^{01}=
-  \begin{subarray}{l}
-  \hphantom{[}\begin{array}{ccc}\leftarrow&n&\rightarrow\end{array}\hphantom{\Bigg]}\\
-  \left[
-    \begin{array}{ccc}
-      q&\cdots&q\\
-      0&\cdots&0\\
-      \vdots&\ddots&\vdots\\
-      0&\cdots&0
-    \end{array}
-  \right]\begin{array}{c}
-    \\\uparrow\\m-1\\\downarrow
-  \end{array}
-\end{subarray}
-  &&
-  R^{01}
-  =\begin{bmatrix}
-    r_{01}&\cdots&r_{01}\\
-    0&\cdots&0\\
-    \vdots&\ddots&\vdots\\
-    0&\cdots&0
-  \end{bmatrix}
-  &&
-  R^{10}
-  =\begin{bmatrix}
-    r_{10}&\cdots&r_{10}\\
-    0&\cdots&0\\
-    \vdots&\ddots&\vdots\\
-    0&\cdots&0
-  \end{bmatrix}
-  &&
-  D^{01}
-  =\begin{bmatrix}
-    d_{01}&\cdots&d_{01}\\
-    0&\cdots&0\\
-    \vdots&\ddots&\vdots\\
-    0&\cdots&0
-  \end{bmatrix}
-\end{align}
-where only the first row is nonzero. The other entries, which correspond to the
-completely uncorrelated replicas in an \textsc{rsb} picture, are all zero
-because uncorrelated vectors on the sphere are orthogonal.
 
-The inverse of block hierarchical matrix is still a block hierarchical matrix, since
+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}
-  \begin{bmatrix}
-    C^{00}&iR^{00}\\iR^{00}&D^{00}
-  \end{bmatrix}^{-1}
-  =
-  \begin{bmatrix}
-    (C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00} & -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} \\
-    -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} & (C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}
-  \end{bmatrix}
+  q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N
+  =\frac{x_0}{1-q}
 \end{equation}
-Because of the structure of the 01 matrices, the volume element will depend
-only on the diagonals of the matrices in this inverse block matrix. If we define
-\begin{align}
-  \tilde c_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}]_{\text d} \\
-  \tilde r_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00}]_{\text d} \\
-  \tilde d_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00}]_{\text d}
-\end{align}
-as the diagonals of the blocks of the inverse matrix, then the result  of the product is
-\begin{equation}
-  \begin{aligned}
-     &   \begin{bmatrix}
-       C^{01}&iR^{01}\\iR^{10}&D^{01}
-     \end{bmatrix}^T
-     \begin{bmatrix}
-       C^{00}&iR^{00}\\iR^{00}&D^{00}
-     \end{bmatrix}^{-1}
-     \begin{bmatrix}
-       C^{01}&iR^{01}\\iR^{10}&D^{01}
-     \end{bmatrix} \\
-     &\qquad=\begin{bmatrix}
-       q^2\tilde d_\mathrm d^{00}+2qr_{10}\tilde r^{00}_\mathrm d-r_{10}^2\tilde d^{00}_\mathrm d
-      &
-      i\left[d_{01}(r_{10}\tilde c^{00}_\mathrm d-q\tilde r^{00}_\mathrm d)+r_{01}(r_{10}\tilde r^{00}_\mathrm d+q\tilde d^{00}_\mathrm d)\right]
-      \\
-      i\left[d_{01}(r_{10}\tilde c^{00}_\mathrm d-q\tilde r^{00}_\mathrm d)+r_{01}(r_{10}\tilde r^{00}_\mathrm d+q\tilde d^{00}_\mathrm d)\right]
-      &
-      d_{01}^2\tilde c^{00}_\mathrm d+2r_{01}d_{01}\tilde r^{00}_\mathrm d-r_{01}^2\tilde d^{00}_\mathrm d
-     \end{bmatrix}
-  \end{aligned}
-\end{equation}
-where each block is a constant $n\times n$ matrix. Because the matrices
-$C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in the replica symmetric case,
-the diagonals of the blocks above take a simple form:
-\begin{align}
-  \tilde c_\mathrm d^{00}=f'(1) &&
-  \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) &&
-  \tilde d_\mathrm d^{00}=d^{00}_\mathrm df'(1)
-\end{align}
-Once these expressions are inserted into the complexity, the limits of $n$ and
-$m$ to zero can be taken, and the parameters from $D^{01}$ and $D^{11}$ can be
-extremized explicitly. The resulting expression for the complexity, which must
-still be extremized over the parameters $\hat\beta_1$, $r^{01}$,
-$r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is
-\begin{equation}
-  \begin{aligned}
-    &\Sigma_{12}(E_0,\mu_0,E_1,\mu_1,q)
-    =\mathop{\mathrm{extremum}}_{\hat\beta_1,r^{11}_\mathrm d,r^{11}_0,r^{01},q^{11}_0}\Bigg\{
-    \mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1
-    +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)\\
-    &\qquad+\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q)
-    +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)\\
-    &+\frac12\Bigg[
-      \hat\beta_1^2\big(f(1)-f(q^{11}_{0})\big)
-      +(r^{11}_\mathrm d)^2f''(1)+2r^{01}r^{10}f''(q)-(r^{11}_0)^2f''(q^{11}_0)
-      +\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1)
-      \\
-    &\qquad+\frac{1-q^2}{1-q^{11}_0}+\left(
-      (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}
-      \left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right)
-    \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \\
-    &\qquad
-    -\frac1{f'(1)}\frac{f'(1)^2-f'(q)^2}{f'(1)-f'(q^{11}_0)}
-    +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)
-    +\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right)
-  \Bigg]\Bigg\}
-  \end{aligned}
-\end{equation}
-It is possible to further extremize this expression over all the other
-variables but $q_0^{11}$, for which the saddle point conditions have a unique
-solution. However, the resulting expression is quite complicated and provides
-no insight. In practice, the complexity can be calculated in two ways. First,
-the extremal problem can be done numerically, initializing from $q=0$ where the
-problem reduces to that of the single-point complexity of points with energy
-$E_1$ and stability $\mu_1$, and then taking small steps in $q$ or other
-parameters to trace out the solution. This is how the data in all the plots of
-this paper was produced. Second, the complexity can be calculated in the near
-neighborhood of a reference point by expanding in small $1-q$. This is what we
-describe in the next subsection.
+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}.
 
-\subsection{Expansion in the near neighborhood}
+\begin{figure}
+  \includegraphics{figs/isolated_eigenvalue.pdf}
+  \hfill
+  \includegraphics{figs/eigenvector_overlap.pdf}
 
-If there is no overlap gap between the reference point and its nearest
-neighbors, their complexity can be calculated by an expansion in $1-q$. First,
-we'll use this method to describe the most common type of stationary point in
-the close vicinity of a reference point. The most common neighbors of a
-reference point are given by further maximizing the two-point complexity over
-the energy $E_1$ and stability $\mu_1$ of the nearby points. This gives the
-conditions
-\begin{align}
-  \hat\beta_1=0 &&
-  \mu_1=2r^{11}_\mathrm df''(1)
-\end{align}
-where the second is only true for $\mu_1^2\leq\mu_\mathrm m^2$, i.e., when the
-nearby points are saddle points or marginal minima. When these conditions are
-inserted into the complexity, an expansion is made in small $1-q$, and the
-saddle point in the remaining parameters is taken, the result is
-\begin{equation}
-  \Sigma_{12}
-  =\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)\big(f''(1)-f'(1)\big)}{f'''(1)f'(1)}}-1\right)(1-q)
-  +O\big((1-q)^2\big)
-\end{equation}
-independent of $E_0$. To describe the properties of these most common
-neighbors, it is convenient to first make a definition. The population of
-stationary points that are most common at each energy (the blue line in
-Fig.~\ref{fig:complexities}) have the relation
-\begin{equation}
-  E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0
-\end{equation}
-between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this
-definition, the energy and stability of the most common neighbors at small
-$\Delta q$ are
-\begin{align} \label{eq:expansion.E.1}
-  E_1&=E_0+\frac12\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\
-    \label{eq:expansion.mu.1}
-  \mu_1&=\mu_0-\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big)
-\end{align}
-The most common neighboring saddles to a reference saddle are much nearer to
-the reference in energy ($\Delta q^2$) than in stability ($\Delta q$). In fact,
-this scaling also holds for the entire range of neighbors to a reference
-saddle, with the limits in energy scaling like $\Delta q^2$ and those of
-stability scaling like $\Delta q$.
+  \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}
 
-Because both expressions are proportional to $E_0-E_\mathrm{dom}(\mu_0)$,
-whether the energy and stability of nearby points increases or decreases from
-that of the reference point depends only on whether the energy of the reference
-point is above or below that of the most common population at the same
-stability. In particular, since $E_\mathrm{dom}(\mu_\mathrm m)=E_\mathrm{th}$,
-the threshold energy is also the pivot around which the points asymptotically
-nearby marginal minima change their properties.
+\section{Conclusion}
+\label{sec:conclusion}
 
-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 $\Delta q$ that no stationary points are
-point any nearer. For the marginal minima, it is not clear that the same should be true.
+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.
 
-When $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion is
+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_{12}
-  =\frac12\frac{f'''(1)v_f}{f''(1)^{3/2}u_f}
-  \left(\sqrt{\frac{2\big[f'(1)(f'''(1)-f''(1))+f''(1)^2\big]}{f'(1)f'''(1)}}-1\right)\big(E_0-E_\textrm{th}\big)(1-q)^2+O\big((1-q)^3\big)
+  \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}
-Note that this expression is only true for $\mu=\mu_\mathrm m$. Therefore,
-among marginal minima, when $E_0$ is greater than the threshold one finds
-neighbors at arbitrarily close distance. When $E_0$ is less than the threshold,
-the complexity of nearby points is negative, and there is a desert where none
-are found.
+which puts the two points on equal footing. This calculation and exploration of
+the atypical neighborhoods it reveals is a clear future direction.
 
-\begin{figure}
-  \centering
-  \includegraphics{figs/expansion_energy.pdf}
-  \hspace{1em}
-  \includegraphics{figs/expansion_stability.pdf}
+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.
 
-  \caption{
-    Demonstration of the convergence of the $(1-q)$-expansion for marginal
-    reference minima. Solid lines and shaded region show are the same as in
-    Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$.
-    The dotted lines show the expansion of most common neighbors, while the
-    dashed lines in both plots show the expansion for the minimum and maximum
-    energies and stabilities found at given $q$.
-  } \label{fig:expansion}
-\end{figure}
+\paragraph{Acknowledgements}
 
-The properties of the nearby states above the threshold can be
-further quantified. The most common points are still given by
-\eqref{eq:expansion.E.1} and \eqref{eq:expansion.mu.1}, but the range of
-available points can also be computed, and one finds that the stability lies in
-the range
-$\mu_1=\mu_\mathrm m+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$
-where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1}
-and
+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 two-point complexity}
+\label{sec:complexity-details}
+
+\subsection{The Hessian factors}
+
+The double partial derivatives of the energy are Gaussian with the variance
 \begin{equation}
-  \delta\mu_2=\frac{v_f}{f'(1)f''(1)^{3/4}}\sqrt{
-    \frac{E_0-E_\mathrm{th}}2\frac{2f''(1)\big(f''(1)-f'(1)\big)+f'(1)f'''(1)}{u_f}
-  }
+  \overline{(\partial_i\partial_jH(\mathbf s))^2}=\frac1Nf''(1)
 \end{equation}
-Similarly, one finds that the energy lies in the range $E_1=E_0+\delta
-E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given
-by the coefficient in \eqref{eq:expansion.E.1} and
+which means that the matrix of partial derivatives belongs to the GOE class. Its spectrum is given by the Wigner semicircle
 \begin{equation}
-  \begin{aligned}
-    \delta E_2
-    &=\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\bigg(
-      \frac{
-        v_f
-        }{3u_f}
-        \big[
-          f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))-2f''(1)^2
-        \big]
-        \\
-    &\hspace{12pc}\times
-        \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)-6f''(1)^2
-        \big]
-    \bigg)^\frac12
-  \end{aligned}
+  \rho(\lambda)=\begin{cases}
+    \frac2{\pi}\sqrt{1-\big(\frac{\lambda}{\mu_\text m}\big)^2} & \lambda^2\leq\mu_\text m^2 \\
+    0 & \text{otherwise}
+  \end{cases}
 \end{equation}
-and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic 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 Wigner semicircle shifted
+by $\omega$, or $\rho(\lambda+\omega)$.
+
+The average over factors depending on the Hessian alone can be made separately
+from those depending on the gradient or energy, since for random Gaussian
+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}. At leading order, the various expectations factorize, each yielding
 \begin{equation}
-  0=((16-(6-\delta q_0)\delta q_0)\delta q_0-12)f'(1)f'''(1)-2\delta q_0(f''(1)-f'(1))f''(1)
+  \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}
-These predictions from the small $1-q$ expansion are compared with numeric
-saddle points for the complexity of marginal minima in
-Fig.~\ref{fig:expansion}, and the results agree well at small $1-q$.
-
-
-\section{Isolated eigenvalue}
-\label{sec:eigenvalue}
-
-The two-point complexity depends on the spectrum at both stationary points
-through the determinant of their Hessians, but only on the bulk of the
-distribution. As we saw, this bulk is unaffected by the conditions of energy
-and proximity. However, these conditions give rise to small-rank perturbations
-to the Hessian, which can lead a subextensive number of eigenvalues leaving the
-bulk. We study the possibility of \emph{one} stray eigenvalue.
-
-We use a technique recently developed to find the smallest eigenvalue of a
-random matrix \cite{Ikeda_2023_Bose-Einstein-like}. One defines a quadratic
-statistical mechanics model with configurations defined on the sphere, whose
-interaction tensor is given by the matrix of interest. By construction, the
-ground state is located in the direction of the eigenvector associated with the
-smallest eigenvalue, and the ground state energy is proportional to that
-eigenvalue.
-
-\begin{figure}
-  \centering
-  \begin{tikzpicture}
-    \def\R{4 } % sphere radius
-    \def\Rt{2 } % tangent plane radius
-    \def\angEl{15} % elevation angle
-    \def\angsa{-160} % azimuth of s_1
-    \def\angq{40} % elevation of constraint circle
-    \filldraw[ball color=white] (0,0) circle (\R);
-  %  \filldraw[fill=white] (0,0) circle (\R);
-
-    \foreach \t in {0,\angq} { \DrawLatitudeCircle[\R]{\t} }
-    %\foreach \t in {\angsa} { \DrawLongitudeCircle[\R]{\t} }
-
-    \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
-    \coordinate (O) at (0,0);
-    \node[circle,draw,black,scale=0.3] at (0,0) {};
-    \coordinate (N) at (0,\H);
-    \draw node[right=10,below] at (0,\H){$\pmb\sigma_1$};
-    \draw[thick, ->](O)--(N);
-
-    \NewLatitudePlane[planeP]{\R}{\angEl}{\angq};
-    \path[planeP] (\angsa:\R) coordinate (P);
-    \path[planeP] (0:1.5*\R) coordinate (Q);
-    \path[planeP] (0:\R) coordinate (Q2);
-    \draw[left] node at (P){$\mathbf s_1$};
-
-    \NewLatitudePlane[equator]{\R}{\angEl}{00};
-    \path[equator] (-30:\R) coordinate (Pprime);
-    \path[equator] (0:{1.5*cos(\angq)*\R}) coordinate (Qe);
-    \path[equator] (0:\R) coordinate (Qe2);
-    \draw node[right=5,below] at (Pprime){$\pmb\sigma_c$};
-
-    \NewLatitudePlane[sbplane]{\R}{\angEl}{\angq};
-    \path[sbplane] (20:\R) coordinate (sb);
-    \draw node[right=3,above=1] at (sb){$\mathbf s_b$};
-
-    \TangentPlane[tplane]{\R}{\angEl}{\angq}{\angsa};
-    \draw[tplane,fill=gray,fill opacity=0.3] circle (\Rt);
-    \draw[tplane,->,thick] (0,0) -> ({\Rt*cos(160)},{\Rt*sin(160)}) node[above=1.5,right] {$\mathbf x_a$};
-    \draw[tplane,->,thick] (0,0) -> ({\Rt*cos(250)},{\Rt*sin(250)}) node[above=1.5,left=0.1] {$\mathbf x_b$};
-
-    \draw[thick, ->] (O)->(P);
-    \draw[thick, ->] (O)->(Pprime);
-    \draw[thick, ->] (O)->(sb);
-
-    \draw[dotted] (Qe) -- (Qe2);
-    \draw[dotted] (Q2) -- (Q);
-    \draw[decorate, decoration = {brace,raise=3}] (Q) -- (Qe) node[pos=0.5,right=7]{$q$};
-  \end{tikzpicture}
-  \caption{
-    A sketch of the vectors involved in the calculation of the isolated
-    eigenvalue. All replicas $\mathbf x$, which correspond with candidate
-    eigenvectors of the Hessian evaluated at $\mathbf s_1$, sit in an $N-2$
-    sphere corresponding with the tangent plane (not to scale) of the first
-    $\mathbf s$ replica. All of the $\mathbf s$ replicas lie on the sphere,
-    constrained to be at fixed overlap $q$ with the first of the $\pmb\sigma$
-    replicas, the reference configuration. All of the $\pmb\sigma$ replicas lie
-    on the sphere.
-  } \label{fig:sphere}
-\end{figure}
-
-Our matrix of interest is the Hessian evaluated at a stationary point of the mixed spherical
-model, conditioned on the relative position, energies, and stabilities
-discussed above. We must restrict the artificial spherical model to lie in the
-tangent plane of the `real' spherical configuration space at the point of
-interest, to avoid our eigenvector pointing in a direction that violates the
-spherical constraint. A sketch of the setup is shown in Fig.~\ref{fig:sphere}. The free energy of this model given a point $\mathbf s$
-and a specific realization of the disordered Hamiltonian is
+Therefore, all of the Lagrange multipliers are fixed to the stabilities $\mu$. We define the function
 \begin{equation}
   \begin{aligned}
-    \beta F_H(\beta\mid\mathbf s,\omega)
-    &=-\frac1N\log\left(\int d\mathbf x\,\delta(\mathbf x\cdot\mathbf s)\delta(\|\mathbf x\|^2-N)\exp\left\{
-        -\beta\frac12\mathbf x^T\operatorname{Hess}H(\mathbf s,\omega)\mathbf x
-    \right\}\right) \\
-    &=-\lim_{\ell\to0}\frac1N\frac\partial{\partial\ell}\int\left[\prod_{\alpha=1}^\ell d\mathbf x_\alpha\,\delta(\mathbf x_\alpha^T\mathbf s)\delta(N-\mathbf x_\alpha^T\mathbf x_\alpha)\exp\left\{
-      -\beta\frac12\mathbf x^T_\alpha\big(\partial\partial H(\mathbf s)+\omega I\big)\mathbf x_\alpha
-    \right\}\right]
+    \mathcal D(\mu)
+    &=\int d\lambda\,\rho(\lambda+\mu)\log|\lambda| \\
+    &=\begin{cases}
+      \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
+       & \mu^2\leq\mu_\text m^2 \\
+      \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2}
+      -\left|\frac{\mu}{\mu_\text m}\right|\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}
+      -\log\left(\left|\frac{\mu}{\mu_\text m}\right|-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu^2>\mu_\text m^2
+    \end{cases}
   \end{aligned}
 \end{equation}
-where the first $\delta$-function keeps the configurations in the tangent
-plane, and the second enforces the spherical constraint. We have anticipated
-treating the logarithm with replicas. We are interested in points $\mathbf s$
-that have certain properties: they are stationary points of $H$ with given
-energy density and stability, and fixed overlap from a reference configuration
-$\pmb\sigma$. We therefore average the free energy above over such points,
-giving
+and the full factor due to the Hessians is
 \begin{equation}
-  \begin{aligned}
-    F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma)
-    &=\int\frac{d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s)}{\int d\nu_H(\mathbf s',\omega'\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s')}F_H(\beta\mid\mathbf s,\omega) \\
-    &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F_H(\beta\mid\mathbf s_1,\omega_1)
-  \end{aligned}
+  e^{Nm\mathcal D(\mu_0)+Nn\mathcal D(\mu_1)}\left[\prod_a^m\delta(N\mu_0-N\varsigma_a)\right]\left[\prod_a^n\delta(N\mu_1-N\omega_a)\right]
 \end{equation}
-again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability. Averaging over these conditions gives
+
+\subsection{The other factors}
+
+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}
-  \begin{aligned}
-    F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)
-    &=\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma) \\
-    &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu_H(\pmb\sigma_a,\varsigma_a\mid E_0,\mu_0)\right]\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma_1)
-  \end{aligned}
+  e^{
+    Nm\hat\beta_0E_0+Nn\hat\beta_1E_1
+    -\sum_a^m\left[(\pmb\sigma_a\cdot\hat{\pmb\sigma}_a)\mu_0
+      -\frac12\hat\mu_0(N-\pmb\sigma_a\cdot\pmb\sigma_a)
+    \right]
+      -\sum_a^n\left[(\mathbf s_a\cdot\hat{\mathbf s}_a)\mu_1
+      -\frac12\hat\mu_1(N-\mathbf s_a\cdot\mathbf s_a)
+      -\frac12\hat\mu_{12}(Nq-\pmb\sigma_1\cdot\mathbf s_a)
+    \right]
+    +\int d\mathbf t\,\mathcal O(\mathbf t)H(\mathbf t)
+  }
 \end{equation}
-This formidable expression is now ready to be averaged over the disordered Hamiltonians $H$. Once averaged,
-the minimum eigenvalue of the conditioned Hessian is then given by twice the ground state energy, or
+where we have introduced the linear operator
 \begin{equation}
-  \lambda_\text{min}=2\lim_{\beta\to\infty}\overline{F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)}
+  \mathcal O(\mathbf t)
+  =\sum_a^m\delta(\mathbf t-\pmb\sigma_a)\left(
+    i\hat{\pmb\sigma}_a\cdot\partial_{\mathbf t}-\hat\beta_0
+  \right)
+  +
+  \sum_a^n\delta(\mathbf t-\mathbf s_a)\left(
+    i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}-\hat\beta_1
+  \right)
 \end{equation}
-For this calculation, there are three different sets of replicated variables.
-Note that, as for the computation of the complexity, the $\pmb\sigma_1$ and
-$\mathbf s_1$ replicas are \emph{special}. The first again is the only of the
-$\pmb\sigma$ 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
+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 $H$-dependent terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that
 \begin{equation}
-  q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N
-  =\frac{x_0}{1-q}
+  \overline{e^{\int d\mathbf t\,\mathcal O(\mathbf t)H(\mathbf t)}}
+  =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}
-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}
+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 are standard.
 
-\section{Conclusion}
-\label{sec:conclusion}
+\subsection{Hubbard--Stratonovich}
 
-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.
+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 &&
+    R^{00}_{ab}=-i\frac1N{\pmb\sigma}_a\cdot\hat{\pmb\sigma}_b &&
+    D^{00}_{ab}=\frac1N\hat{\pmb\sigma}_a\cdot\hat{\pmb\sigma}_b \\
+    C^{01}_{ab}=\frac1N\pmb\sigma_a\cdot\mathbf s_b &&
+    R^{01}_{ab}=-i\frac1N{\pmb\sigma}_a\cdot\hat{\mathbf s}_b &&
+    R^{10}_{ab}=-i\frac1N\hat{\pmb\sigma}_a\cdot{\mathbf s}_b &&
+    D^{01}_{ab}=\frac1N\hat{\pmb\sigma}_a\cdot\hat{\mathbf s}_b \\
+    C^{11}_{ab}=\frac1N\mathbf s_a\cdot\mathbf s_b &&
+    R^{11}_{ab}=-i\frac1N{\mathbf s}_a\cdot\hat{\mathbf s}_b &&
+    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
+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
+can be integrated over. Conjugate matrix field integrals created when the
+$\delta$-functions are promoted to exponentials can be evaluated by saddle
+point in the standard way, yielding an effective action depending on the above
+matrix fields alone.
 
-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.
+\subsection{Saddle point}
 
-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.
+We will always assume that the square matrices $C^{00}$, $R^{00}$, $D^{00}$,
+$C^{11}$, $R^{11}$, and $D^{11}$ are hierarchical matrices, with each set of
+three sharing the same hierarchical structure. In particular, we immediately
+define $c_\mathrm d^{00}$, $r_\mathrm d^{00}$, $d_\mathrm d^{00}$, $c_\mathrm d^{11}$, $r_\mathrm d^{11}$, and
+$d_\mathrm d^{11}$ as the value of the diagonal elements of these matrices,
+respectively. Note that $c_\mathrm d^{00}=c_\mathrm d^{11}=1$ due to the spherical constraint.
 
-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.,
+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}
-  \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)}
+  \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_{11},\mathcal Q_{01}\mid\mathcal Q_{00})}
 \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}
+where
+\begin{equation} \label{eq:one-point.action}
+  \begin{aligned}
+    &\mathcal S_0(\mathcal Q_{00})
+    =\hat\beta_0E_0-r^{00}_\mathrm d\mu_0-\frac12\hat\mu_0(1-c^{00}_\mathrm d)+\mathcal D(\mu_0)\\
+    &\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}
+\bigg\}
+\end{aligned}
+\end{equation}
+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_{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})
+    \right]\right\}
+    \\
+    &\quad+\frac1n\bigg\{
+      \frac12\sum_{ab}^n\left[
+        \hat\beta_1^2f(C^{11}_{ab})+(2\hat\beta_1R^{11}_{ab}-D^{11}_{ab})f'(C^{11}_{ab})+(R^{11}_{ab})^2f''(C^{11}_{ab})
+      \right]\\
+    &\quad+\frac12\log\det\left(
+      \begin{bmatrix}
+        C^{11}&iR^{11}\\iR^{11}&D^{11}
+      \end{bmatrix}-
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\iR^{10}&D^{01}
+      \end{bmatrix}^T
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\iR^{10}&D^{01}
+      \end{bmatrix}
+    \right)
+  \bigg\}
+  \end{aligned}
+\end{equation}
+Because of the structure of this problem in the twin limits of $m$ and $n$ to
+zero, the parameters $\mathcal Q_{00}$ can be evaluated at a saddle point of
+$\mathcal S_0$ alone. This means that these parameters will take the same value
+they take when the ordinary, 1-point complexity is calculated. For a replica
+symmetric complexity of the reference point, this results in
+\begin{align}
+  \hat\beta_0
+  &=-\frac{\mu_0f'(1)+E_0\big(f'(1)+f''(1)\big)}{u_f}\\
+  r_\mathrm d^{00}
+  &=\frac{\mu_0f(1)+E_0f'(1)}{u_f} \\
+  d_\mathrm d^{00}
+  &=\frac1{f'(1)}
+  -\left(
+    \frac{\mu_0f(1)+E_0f'(1)}{u_f}
+  \right)^2
+\end{align}
+where we define for brevity (here and elsewhere) the constants
+\begin{align}
+  u_f=f(1)\big(f'(1)+f''(1)\big)-f'(1)^2
+  &&
+  v_f=f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2
+\end{align}
+Note that because the coefficients of $f$ must be nonnegative for $f$ to
+be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. Note also
+that $u_f=v_f=0$ if $f$ is a homogeneous polynomial as in the pure models.
+These expressions are invalid for the pure models because $\mu_0$ and $E_0$
+cannot be fixed independently; we would have done the equivalent of inserting
+two identical $\delta$-functions. For the pure models, the terms $\hat\beta_0$ and
+$\hat\beta_1$ must be set to zero in our prior formulae (as if the energy was
+not constrained) and then the saddle point taken.
 
-JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the
-INFN.
 
-\appendix
+In general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$,
+and $D^{01}$ to have constant \emph{rows} of length $n$, with blocks of rows
+corresponding to the \textsc{rsb} structure of the single-point complexity. For
+the scope of this paper, where we restrict ourselves to replica symmetric
+complexities, they have the following form at the saddle point:
+\begin{align} \label{eq:01.ansatz}
+  C^{01}=
+  \begin{subarray}{l}
+  \hphantom{[}\begin{array}{ccc}\leftarrow&n&\rightarrow\end{array}\hphantom{\Bigg]}\\
+  \left[
+    \begin{array}{ccc}
+      q&\cdots&q\\
+      0&\cdots&0\\
+      \vdots&\ddots&\vdots\\
+      0&\cdots&0
+    \end{array}
+  \right]\begin{array}{c}
+    \\\uparrow\\m-1\\\downarrow
+  \end{array}
+\end{subarray}
+  &&
+  R^{01}
+  =\begin{bmatrix}
+    r_{01}&\cdots&r_{01}\\
+    0&\cdots&0\\
+    \vdots&\ddots&\vdots\\
+    0&\cdots&0
+  \end{bmatrix}
+  &&
+  R^{10}
+  =\begin{bmatrix}
+    r_{10}&\cdots&r_{10}\\
+    0&\cdots&0\\
+    \vdots&\ddots&\vdots\\
+    0&\cdots&0
+  \end{bmatrix}
+  &&
+  D^{01}
+  =\begin{bmatrix}
+    d_{01}&\cdots&d_{01}\\
+    0&\cdots&0\\
+    \vdots&\ddots&\vdots\\
+    0&\cdots&0
+  \end{bmatrix}
+\end{align}
+where only the first row is nonzero. The other entries, which correspond to the
+completely uncorrelated replicas in an \textsc{rsb} picture, are all zero
+because uncorrelated vectors on the sphere are orthogonal.
 
+The inverse of block hierarchical matrix is still a block hierarchical matrix, since
+\begin{equation}
+  \begin{bmatrix}
+    C^{00}&iR^{00}\\iR^{00}&D^{00}
+  \end{bmatrix}^{-1}
+  =
+  \begin{bmatrix}
+    (C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00} & -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} \\
+    -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} & (C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}
+  \end{bmatrix}
+\end{equation}
+Because of the structure of the 01 matrices, the volume element will depend
+only on the diagonals of the matrices in this inverse block matrix. If we define
+\begin{align}
+  \tilde c_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}]_{\text d} \\
+  \tilde r_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00}]_{\text d} \\
+  \tilde d_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00}]_{\text d}
+\end{align}
+as the diagonals of the blocks of the inverse matrix, then the result  of the product is
+\begin{equation}
+  \begin{aligned}
+     &   \begin{bmatrix}
+       C^{01}&iR^{01}\\iR^{10}&D^{01}
+     \end{bmatrix}^T
+     \begin{bmatrix}
+       C^{00}&iR^{00}\\iR^{00}&D^{00}
+     \end{bmatrix}^{-1}
+     \begin{bmatrix}
+       C^{01}&iR^{01}\\iR^{10}&D^{01}
+     \end{bmatrix} \\
+     &\qquad=\begin{bmatrix}
+       q^2\tilde d_\mathrm d^{00}+2qr_{10}\tilde r^{00}_\mathrm d-r_{10}^2\tilde d^{00}_\mathrm d
+      &
+      i\left[d_{01}(r_{10}\tilde c^{00}_\mathrm d-q\tilde r^{00}_\mathrm d)+r_{01}(r_{10}\tilde r^{00}_\mathrm d+q\tilde d^{00}_\mathrm d)\right]
+      \\
+      i\left[d_{01}(r_{10}\tilde c^{00}_\mathrm d-q\tilde r^{00}_\mathrm d)+r_{01}(r_{10}\tilde r^{00}_\mathrm d+q\tilde d^{00}_\mathrm d)\right]
+      &
+      d_{01}^2\tilde c^{00}_\mathrm d+2r_{01}d_{01}\tilde r^{00}_\mathrm d-r_{01}^2\tilde d^{00}_\mathrm d
+     \end{bmatrix}
+  \end{aligned}
+\end{equation}
+where each block is a constant $n\times n$ matrix. Because the matrices
+$C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in the replica symmetric case,
+the diagonals of the blocks above take a simple form:
+\begin{align}
+  \tilde c_\mathrm d^{00}=f'(1) &&
+  \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) &&
+  \tilde d_\mathrm d^{00}=d^{00}_\mathrm df'(1)
+\end{align}
+Once these expressions are inserted into the complexity, the limits of $n$ and
+$m$ to zero can be taken, and the parameters from $D^{01}$ and $D^{11}$ can be
+extremized explicitly. 
 \section{Details of calculation for the isolated eigenvalue}
 \label{sec:eigenvalue-details}
 
-- 
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