path: root/2-point.tex

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\begin{document}

\title{
  Arrangement of nearby minima and saddles in the mixed spherical energy landscapes
}

\author{Jaron Kent-Dobias}
\affil{Istituto Nazionale di Fisica Nucleare, Sezione di Roma I}

\maketitle
\begin{abstract}
  The mixed spherical models were recently found to violate long-held
  assumptions about mean-field glassy dynamics. In particular, the threshold
  energy, where most stationary points are marginal and which in the simpler
  pure models attracts long-time dynamics, seems to lose significance. Here,
  we compute the typical distribution of stationary points relative to each
  other in mixed models with a replica symmetric complexity. We examine the
  stability of nearby points, accounting for the presence of an isolated
  eigenvalue in their spectrum due to their proximity. Despite finding rich
  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 of most marginal inherent states into a continuous manifold.
\end{abstract}

\tableofcontents

\section{Introduction}

Many systems exhibit ``glassiness,'' characterized by rapid slowing of dynamics
over a short parameter interval. These include actual (structural) glasses,
spin glasses, certain inference and optimization problems, and more. Glassiness
is qualitatively understood to arise from structure of an energy or cost
landscape, whether due to the proliferation of metastable states, or to the
raising of barriers which cause effective dynamic constraints
\cite{Cavagna_2001_Fragile, Stillinger_2013_Glass,
Kirkpatrick_2015_Colloquium}. However, in most models there is no known
quantitative correspondence between these landscape properties and the dynamic
behavior they are purported to describe.

There is such a correspondence in one of the simplest mean-field model of
glasses: in the pure spherical models, the dynamic transition corresponds with
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
ways: it attracts the long-time dynamics after quenches in temperature to below
the dynamical transition from any starting temperature
\cite{Biroli_1999_Dynamical, Sellke_2023_The}. All of this can be understood in terms of the
landscape structure.

In slightly less simple models, the mixed spherical models, the story changes.
There are now a range of energies with exponentially many marginal minima. It
was believed that the energy level at which these marginal minima are the most
common type of stationary point would play the same role as the threshold
energy in the pure models (in fact we will refer to this energy level as the
threshold energy in the mixed models). However, recent work has shown that
this is incorrect. Quenches from different starting temperatures above the
dynamical transition temperature result in dynamics that approach different
energy levels, and the purported threshold does not attract the long-time
dynamics in most cases \cite{Folena_2020_Rethinking, Folena_2021_Gradient}.

This paper studies the two-point structure of stationary points in the mixed
spherical models, or their arrangement relative to each other, previously
studied only for the pure models \cite{Ros_2019_Complexity}. This gives various
kinds of information. When one point is a minimum, we see what other kinds of
minima are nearby, and what kind of saddle points (barriers) separate them.
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 constrained points 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 inherent states separated by subextensive barriers \cite{Kurchan_1996_Phase} is only true
at the threshold energy, while at other energies typical marginal minima are far apart
and separated by extensive barriers. Therefore, with respect to the problem of
dynamics this paper merely deepens the outstanding issues.

In \S\ref{sec:model}, we introduce the mixed spherical models and discuss their
properties, defining many of the symbols and concepts relied on in the rest of
the paper. In \S\ref{sec:results}, we share the main results of the paper, with
a large discussion of the neighborhood of marginal states. 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 an isolated eigenvalue and for the zero-temperature
Franz--Parisi potential. Finally in \S\ref{sec:conclusion} we draw some conclusions from the study.

\section{Model}
\label{sec:model}

The mixed spherical models are defined by the Hamiltonian
\begin{equation} \label{eq:hamiltonian}
  H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
\end{equation}
where the vectors $\mathbf s\in\mathbb R^N$ are confined to the sphere
$\|\mathbf s\|^2=N$ \cite{Kirkpatrick_1987_p-spin-interaction, Crisanti_1992_The, Crisanti_2004_Spherical}.  The coupling coefficients $J$ are fully-connected and random, with
zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ scaled so that
the energy is typically extensive. The overbar denotes an average
over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
constants that define the particular model. For instance, the `pure'
$p$-spin model has $a_{p'}=\delta_{p'p}$. This class of models encompasses all
statistically isotropic Gaussian random Hamiltonians defined on the
hypersphere.

The covariance between the energy at two different points is a function of the overlap, or dot product, between those points, or
\begin{equation} \label{eq:covariance}
  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right)
\end{equation}
where the function $f$ is defined from the coefficients $a_p$ by
\begin{equation}
  f(q)=\frac12\sum_pa_pq^p
\end{equation}
In this paper, we will focus on models with a replica symmetric complexity, but
many of the intermediate formulae are valid for arbitrary replica symmetry
breakings. At most {\oldstylenums1}\textsc{rsb} in the equilibrium is guaranteed if the function
$\chi(q)=f''(q)^{-1/2}$ is convex \cite{Crisanti_1992_The}. The complexity at the ground state must
reflect the structure of equilibrium, and therefore be replica symmetric. We
are not aware of any result guaranteeing this for the complexity away from the
ground state, but we check that our replica-symmetric solutions satisfy the
saddle point equations at {\oldstylenums1}\textsc{rsb}.

To enforce the spherical constraint at stationary points, we make use of a Lagrange multiplier $\omega$. This results in the extremal problem
\begin{equation}
  H(\mathbf s)+\frac\omega2(\|\mathbf s\|^2-N)
\end{equation}
The gradient and Hessian at a stationary point are then
\begin{align}
  \nabla H(\mathbf s,\omega)=\partial H(\mathbf s)+\omega\mathbf s
  &&
  \operatorname{Hess}H(\mathbf s,\omega)=\partial\partial H(\mathbf s)+\omega I
\end{align}
where $\partial=\frac\partial{\partial\mathbf s}$ denotes the derivative with respect to $\mathbf s$.

\begin{figure}
  \includegraphics{figs/spectrum_saddle.pdf}
  \hfill
  \includegraphics{figs/spectrum_marginal.pdf}
  \hfill
  \includegraphics{figs/spectrum_minimum.pdf}\\

  \vspace{1em}

  \includegraphics{figs/spectrum_saddle_2.pdf}
  \hfill
  \includegraphics{figs/spectrum_minimum_2.pdf}
  \hfill
  \includegraphics{figs/spectrum_saddle_3.pdf}

  \caption{
    Illustration of the interpretation of the stability $\mu$, which sets the
    location of the center of the eigenvalue spectrum. In the top row we have
    spectra without an isolated eigenvalue. \textbf{(a)} $\mu<\mu_\mathrm m$,
    there are an extensive number of downward directions, and the associated
    point is an \emph{extensive saddle}. \textbf{(b)} $\mu=\mu_\mathrm m$ and
    we have a \emph{marginal minimum} with asymptotically flat directions.
    \textbf{(c)} $\mu>\mu_\mathrm m$, all eigenvalues are positive, and the
    point is a \emph{stable minimum}. On the bottom we show what happens in the
    presence of an isolated eigenvalue. \textbf{(d)} One eigenvalue leaves the
    bulk spectrum of a saddle point and it remains a saddle point, but now with
    an eigenvector correlated with the orientation of the reference vector, so
    we call this a \emph{oriented saddle}. \textbf{(e)} The same happens for
    a minimum, and we can call it an \emph{oriented minimum}. \textbf{(f)} One
    eigenvalue outside a positive bulk spectrum is negative, destabilizing what
    would otherwise have been a stable minimum, producing an \emph{oriented
    index-one saddle}.
  } \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. 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
  \includegraphics{figs/single_complexity.pdf}
  \caption{
    Plot of the complexity (logarithm of the number of stationary points) for
    the $3+4$ mixed spherical model studied in this paper. Energies and stabilities
    of interest are marked, including the ground state energy and stability
    $E_\mathrm{gs}$, the marginal stability $\mu_\mathrm
    m$, and the threshold energy $E_\mathrm{th}$. The blue line shows the location
    of the most common type of stationary point at each energy level. The
    highlighted red region shows the approximate range of minima which attract
    aging dynamics from a quench to zero temperature found in
    \cite{Folena_2020_Rethinking}.
  } \label{fig:complexities}
\end{figure}

In this study, we focus exclusively on the model studied in
\cite{Folena_2020_Rethinking}, whose covariance function is given by
\begin{equation}
  f_{3+4}(q)=\frac12\big(q^3+q^4\big)
\end{equation}
First, it has convex $f''(q)^{-1/2}$, so at least the ground state complexity
must be replica symmetric, as in the pure spherical models. Second, properties
of its long-time dynamics have been extensively studied. The annealed one-point
complexity of these models was calculated in \cite{BenArous_2019_Geometry}, and
for this model the annealed is expected to be correct. The complexity of this
model is plotted in Fig.~\ref{fig:complexities}.

\section{Results}
\label{sec:results}

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 typical 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}
  \includegraphics{figs/gapped_min_energy.pdf}
  \raisebox{5em}{\includegraphics{figs/gapped_min_energy_legend.pdf}}
  \hfill
  \includegraphics{figs/gapped_min_stability.pdf}
  \raisebox{5em}{\includegraphics{figs/gapped_min_stability_legend.pdf}}

  \caption{
    The neighborhood of a reference minimum with $E_0=-1.71865<E_\mathrm{th}$
    and $\mu_0=6.1>\mu_\mathrm m$. \textbf{Left:} The most common type of
    stationary point lying at fixed overlap $q$ and energy $E_1$ from the
    reference minimum. The black line gives the smallest or largest energies
    where neighbors can be found at a given overlap. \textbf{Right:} The most
    common type of stationary point lying at fixed overlap $q$ and stability
    $\mu_1$ from the reference minimum. Note that this describes a different
    set of stationary points than shown in the left plot. On both plots, the
    shading of the righthand part depicts the state of an isolated eigenvalue
    in the spectrum of the Hessian of the neighboring points. Those more
    lightly shaded are points with an isolated eigenvalue that does not change
    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 lines on both plots depict the
    trajectory of the solid line on the other 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.
  } \label{fig:min.neighborhood}
\end{figure}

If the reference configuration is restricted to stable minima, then there is a
gap in the overlap between those minima and their nearest neighbors in
configuration space. We can characterize these neighbors as a function of their
overlap and stability, with one example seen in
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}.

First, 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 emergence of oriented index-one saddles along
the line of lowest-energy states at a given overlap occurs at the local minimum
of this line. Unlike the pure models, neighbors exist for independent $\mu_1$
and $E_1$, and the line of lowest-energy states at a given overlap is different
from the line of maximally-stable states at a given overlap.

Also like the pure models, there is a correspondence between the maximum of the
zero-temperature Franz--Parisi potential restricted to minima of the specified
type and the local maximum of the neighbor complexity along the line of
lowest-energy states. This is seen in Fig.~\ref{fig:franz-parisi}.

\begin{figure}
  \centering
  \includegraphics{figs/franz_parisi.pdf}

  \caption{
    Comparison of the lowest-energy stationary points at overlap $q$ with a
    reference minimum of $E_0=-1.71865<E_\mathrm{th}$ and
    $\mu_0=6.1>\mu_\mathrm m$ (yellow, top), and the zero-temperature Franz--Parisi potential
    with respect to the same reference minimum (blue, bottom). The two curves
    coincide precisely at their minimum $q=0$ and at the local maximum $q\simeq0.5909$.
  } \label{fig:franz-parisi}
\end{figure}

\subsection{Geometry of marginal states}

The set of marginal states is of special interest. First, it has more structure
than in the pure models, with different types of marginal states being found at
different energies. Second, these states attract the dynamics (as evidenced by power-law relaxations), and so are the
inevitable end-point of equilibrium and algorithmic processes \cite{Folena_2023_On}. We find,
surprisingly, that the properties of marginal states pivot around the threshold
energy, the energy at which most stationary points are marginal.

\begin{itemize}
  \item \textbf{Energies below the threshold.} Marginal states have a
    macroscopic gap in their overlap with nearby minima and saddles. The
    nearest stationary points are saddles with an oriented 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. 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 include oriented saddle
    points with an extensive number of downward directions, and oriented stable
    and 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}

\begin{figure}
  \includegraphics{figs/nearest_energies_below.pdf}
  \hfill
  \includegraphics{figs/nearest_stabilities_below.pdf}
  \hfill
  \includegraphics{figs/nearest_marginal_below.pdf}

  \caption{
    The neighborhood of marginal states at several energies below the threshold
    energy. \textbf{Left:} The range of energies $E_1$ at which nearby states
    are found. For any $E_0<E_\mathrm{th}$, the nearest class of states is at
    an extensive distance, and their energies are higher than that of the
    reference configuration. \textbf{Center:} The range of stabilities $\mu_1$
    at which nearby states are found. For $E_0$ near the threshold, the nearest
    states are always index-one saddles with $\mu>\mu_\mathrm m$, but as the
    overlap gap widens their population becomes model-dependent.
    \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 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{
    The neighborhood of marginal minima at the threshold energy
    $E_0=E_\mathrm{th}$. In all plots, the dashed lines show the population of
    most common neighbors at the given overlap $q$.
    \textbf{Left:} The range of energies $E_1$ at which nearby points are
    found. The approach of both the minimum and maximum energies goes like
    $(1-q)^3$. \textbf{Center:} The range of stabilities $\mu_1$ at which nearby
    points are found. The approach of both limits goes like $(1-q)^2$.
    \textbf{Right:} The range of nearby marginal minima. The more darkly shaded
    region denotes where an isolated eigenvalue appears. Marginal minima at the
    threshold lie asymptotically close together.
  } \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{typical 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, or an atypical population of marginal minima. At energies both below and above the threshold energy,
typical 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
typical marginal states are approached above the threshold energy, they must have been
via the neighborhood of saddles with an extensive index, not other marginal
states. On the other hand, typical marginal states approached below the threshold
energy must  be reached after an extensive distance in configuration space
without encountering any stationary point. The geometric conditions of the
neighborhoods above and below are quite different, but the observed aging
dynamics don't appear to qualitatively change \cite{Folena_2020_Rethinking,
Folena_2021_Gradient}. Therefore, if the marginal minima attracting dynamics are typical, the conditions in the neighborhood of the
marginal minimum eventually reached at infinite time appear to be irrelevant
for the nature of aging dynamics at any finite time.

A version of
this story was told a long time ago by the authors of
\cite{Kurchan_1996_Phase}, who write on aging in the pure spherical models
where the limit of $N\to\infty$ is taken before that of $t\to\infty$: ``it is
important to remark that this [...]\ does \emph{not} mean that the system
relaxes into a near-threshold state: at all finite times an infinite system has
a Hessian with an \emph{infinite} number of directions in which the energy is a
maximum. [...] We have seen that the saddles separating threshold minima are
typically $O(N^{1/3})$ above the threshold level, while the energy is at all
finite times $O(N)$ above this level.'' In the present case of the mixed
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.

On the other hand, it is possible that \emph{atypical} marginal minima are
relevant for attracting the dynamics. Studying these points would require a
different kind of computation, where the fixed reference point is abandoned and
both points are treated on equal footing. Such a calculation is beyond the
scope of this paper, but is clear fodder for future research.

\section{Calculation of the two-point complexity}
\label{sec:complexity}

We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
\begin{equation}
  d\nu_H(\mathbf s,\omega)
  =2\,d\mathbf s\,d\omega\,\delta(\|\mathbf s\|^2-N)\,
  \delta\big(\nabla H(\mathbf s,\omega)\big)\,
  \big|\det\operatorname{Hess}H(\mathbf s,\omega)\big|
\end{equation}
which counts stationary points of the function $H$. If integrated over
configuration space, $\mathcal N_H=\int d\nu_H(\mathbf s,\omega)$ gives the
total number of stationary points in the function. The Kac--Rice method has been used by in many studies to analyze the geometry of random functions \cite{Cavagna_1998_Stationary, Fyodorov_2007_Density, Bray_2007_Statistics}. More interesting is the
measure conditioned on the energy density $E$ and stability $\mu$ of the
points,
\begin{equation}
  d\nu_H(\mathbf s,\omega\mid E,\mu)
  =d\nu_H(\mathbf s,\omega)\,
  \delta\big(NE-H(\mathbf s)\big)\,
  \delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\omega)\big)
\end{equation}
While $\mu$ is strictly the trace of the Hessian, we call it the stability
because in this family of models all stationary points have a bulk spectrum of
the same shape, shifted by different constants. The stability $\mu$ sets this
shift, and therefore determines if the spectrum has bulk support on zero. See
Fig.~\ref{fig:spectra} for examples.

We want the typical number of stationary points with energy density
$E_1$ 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 logarithm of this number. The two-point complexity is
therefore defined by
\begin{equation} \label{eq:complexity.definition}
  \Sigma_{12}
  =\frac1N\overline{\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)}\,
  \log\bigg(\int d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)\bigg)}
\end{equation}
Both the denominator and the logarithm are treated using the replica trick, which yields
\begin{equation}
  \Sigma_{12}
  =\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 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
\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}
\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 $\rho(\lambda+\omega)$.

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}
  \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 to the stabilities $\mu$. We define the function
\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}
\end{equation}
and the full factor due to the Hessians 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]
\end{equation}

\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}
  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}
where we have introduced the linear operator
\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)
\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
\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)}
\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}

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}

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.

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_{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}
  \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.


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. 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.

\subsection{Expansion in the near neighborhood}

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 $\Delta q$.

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}
  \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.

\begin{figure}
  \centering
  \includegraphics{figs/expansion_energy.pdf}
  \hspace{1em}
  \includegraphics{figs/expansion_stability.pdf}

  \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}
  \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}
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}
  \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}
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}
  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}
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
\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]
  \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
\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}
\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
\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}
\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
\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.

Using the same methodology as above, the disorder-dependent terms are captured in the linear operator
\begin{equation}
  \mathcal O(\mathbf t)=
  \sum_a^m\delta(\mathbf t-\pmb\sigma_a)(i\hat{\pmb\sigma}_a\cdot\partial_\mathbf t-\hat\beta_0)
  +
  \sum_b^n\delta(\mathbf t-\mathbf s_b)(i\hat{\mathbf s}_b\cdot\partial_\mathbf t-\hat\beta_1)
  -\frac12
  \delta(\mathbf t-\mathbf s_1)\beta\sum_c^\ell(\mathbf x_c\cdot\partial_{\mathbf t})^2
\end{equation}
that is applied to $H$ by integrating over $\mathbf t\in\mathbb R^N$. The
resulting expression for the integrand produces dependencies only on the
scalar products in \eqref{eq:fields} and on the new scalar products involving
the tangent plane vectors $\mathbf x$,
\begin{align}
  A_{ab}=\frac1N\mathbf x_a\cdot\mathbf x_b
  &&
  X^0_{ab}=\frac1N\pmb\sigma_a\cdot\mathbf x_b
  &&
  \hat X^0_{ab}=-i\frac1N\hat{\pmb\sigma}_a\cdot\mathbf x_b
  &&
  X^1_{ab}=\frac1N\mathbf s_a\cdot\mathbf x_b
  &&
  \hat X^1_{ab}=-i\frac1N\hat{\mathbf s}_a\cdot\mathbf x_b
\end{align}
Defining as before a block variable $\mathcal Q_x=(A,X^0,\hat X^0,X^1,\hat X^1)$
and consolidating the previous block variables $\mathcal Q=(\mathcal Q_{00},
\mathcal Q_{01},\mathcal Q_{11})$, we can write the minimum eigenvalue
schematically as
\begin{equation}
  \lambda_\mathrm{min}
  =-2\lim_{\beta\to\infty}
  \lim_{\substack{\ell\to0\\m\to0\\n\to0}}\frac\partial{\partial\ell}\frac1{\beta N}
  \int d\mathcal Q\,d\mathcal Q_x\,
  e^{N[
    m\mathcal S_0(\mathcal Q_{00})
    +n\mathcal S(\mathcal Q_{11},\mathcal Q_{01}\mid\mathcal Q_{00})
    +\ell\mathcal S_x(\mathcal Q_x\mid\mathcal Q_{00},\mathcal Q_{01},\mathcal Q_{11})
  ]}
\end{equation}
where $\mathcal S_0$ is given by \eqref{eq:one-point.action}, $\mathcal S$ is
given by \eqref{eq:two-point.action}, and the new action $\mathcal S_x$ is
given by
\begin{equation} \label{eq:action.eigenvalue}
  \begin{aligned}
    \ell\mathcal S_x(\mathcal Q_x\mid\mathcal Q)
    =-\frac12\ell\beta\mu+
    \frac12\beta\sum_b^\ell\bigg\{
      \frac12\beta&f''(1)\sum_a^lA_{ab}^2\\
    &+\sum_a^m\left[
        \big(\hat\beta_0f''(C^{01}_{a1})+R^{10}_{a1}f'''(C^{01}_{a1})\big)(X^0_{ab})^2
        +2f''(C^{01}_{a1})X^0_{ab}\hat X^0_{ab}
      \right] \\
    &+\sum_a^n\left[
        \big(\hat\beta_1f''(C^{11}_{a1})+R^{11}_{a1}f'''(C^{11}_{a1})\big)(X^1_{ab})^2
        +2f''(C^{11}_{a1})X^1_{ab}\hat X^1_{ab}
      \right]
    \bigg\}\\
    &+\frac12\log\det\left(
      A-
      \begin{bmatrix}
        X^0\\\hat X^0\\X^1\\\hat X^1
      \end{bmatrix}^T
      \begin{bmatrix}
        C^{00}&iR^{00}&C^{01}&iR^{01}\\
        iR^{00}&D^{00}&iR^{10}&D^{01}\\
        (C^{01})^T&(iR^{10})^T&C^{11}&iR^{11}\\
        (iR^{01})^T&(D^{01})^T&iR^{11}&D^{11}\\
      \end{bmatrix}^{-1}
      \begin{bmatrix}
        X^0\\\hat X^0\\X^1\\\hat X^1
      \end{bmatrix}
    \right)
  \end{aligned}
\end{equation}
As usual in these quenched Franz--Parisi style computations, the saddle point expressions for the variables $\mathcal Q$ in the joint limits of $m$, $n$, and $\ell$ to zero are independent of $\mathcal Q_x$, and so these quantities take the same value they do for the two-point complexity that we computed above. The saddle point conditions for the variables $\mathcal Q_x$ are then fixed by extremizing with respect to the final action.

To evaluate this expression, we need a sensible ansatz for the variables
$\mathcal Q_x$. The matrix $A$ we expect to be an ordinary hierarchical matrix,
and since the model is a spherical 2-spin the finite but low temperature order
will be replica symmetric with nonzero $a_0$. The expected form of the $X$ matrices
follows our reasoning for the 01 matrices of the previous section: namely, they
should have constant rows and a column structure which matches that of the
level of \textsc{rsb} order associated with the degrees of freedom that
parameterize the columns. Since both the reference configurations and the
constrained configurations have replica symmetric order, we expect
\begin{align}
  X^0
  =
  \begin{subarray}{l}
    \hphantom{[}\begin{array}{ccc}\leftarrow&\ell&\rightarrow\end{array}\hphantom{\Bigg]}\\
  \left[
    \begin{array}{ccc}
      x_0&\cdots&x_0\\
      0&\cdots&0\\
      \vdots&\ddots&\vdots\\
      0&\cdots&0
    \end{array}
  \right]\begin{array}{c}
    \\\uparrow\\m-1\\\downarrow
  \end{array}\\
  \vphantom{\begin{array}{c}n\end{array}}
  \end{subarray}
  &&
  \hat X^0
  =
  \left[
    \begin{array}{ccc}
      \hat x_0&\cdots&\hat x_0\\
      0&\cdots&0\\
      \vdots&\ddots&\vdots\\
      0&\cdots&0
    \end{array}
  \right]
  &&
  X^1
  =
  \begin{subarray}{l}
    \hphantom{[}\begin{array}{ccc}\leftarrow&\ell&\rightarrow\end{array}\hphantom{\Bigg]}\\
  \left[
    \begin{array}{ccc}
      0&\cdots&0\\
      x_1&\cdots&x_1\\
      \vdots&\ddots&\vdots\\
      x_1&\cdots&x_1
    \end{array}
  \right]\begin{array}{c}
    \\\uparrow\\n-1\\\downarrow
  \end{array}\\
  \vphantom{\begin{array}{c}n\end{array}}
  \end{subarray}
  &&
  \hat X^1
  =\begin{bmatrix}
    \hat x_1^0&\cdots&\hat x_1^0\\
    \hat x_1^1&\cdots&\hat x_1^1\\
    \vdots&\ddots&\vdots\\
    \hat x_1^1&\cdots&\hat x_1^1
  \end{bmatrix}
\end{align}
Here, the lower block of the 0 matrices is zero, because these replicas have no
overlap with the reference or anything else. The first row of the $X^1$ matrix
needs to be zero because of the constraint that the tangent space vectors lie
in the tangent plane to the sphere, and therefore have $\mathbf x_a\cdot\mathbf
s_1=0$ for any $a$. This produces five parameters to deal with, which we
compile in the vector $\mathcal X=(x_0,\hat x_0,x_1\hat x_1^1,\hat x_1^0)$.

Inserting this ansatz is straightforward in the first part of
\eqref{eq:action.eigenvalue}, but the term with $\log\det$ is more complicated.
We must invert the block matrix inside. We define
\begin{equation}
    \begin{bmatrix}
      C^{00}&iR^{00}&C^{01}&iR^{01}\\
      iR^{00}&D^{00}&iR^{10}&D^{01}\\
      (C^{01})^T&(iR^{10})^T&C^{11}&iR^{11}\\
      (iR^{01})^T&(D^{10})^T&iR^{11}&D^{11}\\
    \end{bmatrix}^{-1}
    =
  \begin{bmatrix}
    M_{11} & M_{12} \\
    M_{12}^T & M_{22}
  \end{bmatrix}
\end{equation}
where the blocks inside the inverse are given by
\begin{align}
  M_{11}&=
    \left(
      \begin{bmatrix}
        C^{00}&iR^{00}\\iR^{00}&D^{00}
      \end{bmatrix}
      -
      \begin{bmatrix}
        C^{01}&iR^{01}\\
        iR^{10}&D^{01}
      \end{bmatrix}
      \begin{bmatrix}
        C^{11}&iR^{11}\\iR^{11}&D^{11}
      \end{bmatrix}^{-1}
      \begin{bmatrix}
        C^{01}&iR^{01}\\
        iR^{10}&D^{01}
      \end{bmatrix}^T
    \right)^{-1}
    \\
  M_{12}&=-
  M_{11}
      \begin{bmatrix}
        C^{01}&iR^{01}\\
        iR^{10}&D^{01}
      \end{bmatrix}
      \begin{bmatrix}
        C^{11}&iR^{11}\\iR^{11}&D^{11}
      \end{bmatrix}^{-1}
      \\
  M_{22}&=
    \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)^{-1}
\end{align}
Here, $M_{22}$ is the inverse of the matrix already analyzed as part of
\eqref{eq:two-point.action}. Following our discussion of the inverses of block
replica matrices above, and reasoning about their products with the rectangular
block constant matrices, things can be worked out from here. For instance, the
second term in $M_{11}$ contributes nothing once the appropriate limits are
taken, because each contribution is proportional to $n$.

The contribution can be written as
\begin{equation} \label{eq:inverse.quadratic.form}
    \begin{bmatrix}
      X_0\\i\hat X_0
    \end{bmatrix}^TM_{11}
    \begin{bmatrix}
      X_0\\i\hat X_0
    \end{bmatrix}
    +
    2\begin{bmatrix}
      X_0\\i\hat X_0
    \end{bmatrix}^TM_{12}
    \begin{bmatrix}
      X_1\\i\hat X_1
    \end{bmatrix}
    +
    \begin{bmatrix}
      X_1\\i\hat X_1
    \end{bmatrix}^TM_{22}
    \begin{bmatrix}
      X_1\\i\hat X_1
    \end{bmatrix}
\end{equation}
and without too much reasoning one can see that the result is an $\ell\times\ell$ constant matrix. If $A$ is a replica matrix and $c$ is a constant, then
\begin{equation}
  \log\det(A-c)=\log\det A-\frac{c}{\sum_{i=0}^k(a_{i+1}-a_i)x_{i+1}}
\end{equation}
where $a_{k+1}=1$ and $x_{k+1}=1$.
The basic form of the action is (for replica symmetric $A$)
\begin{equation}
  2\mathcal S_x(\mathcal Q_x\mid\mathcal Q)
  =-\beta\mu+\frac12\beta^2f''(1)(1-a_0^2)+\log(1-a_0)+\frac{a_0}{1-a_0}+\mathcal X^T\left(\beta B-\frac1{1-a_0}C\right)\mathcal X
\end{equation}
where the matrix $B$ comes from the $\mathcal X$-dependent parts of the first
lines of \eqref{eq:action.eigenvalue} and is given by
\begin{equation}
  B=\begin{bmatrix}
    \hat\beta_0f''(q)+r_{10}f'''(q)&f''(q)&0&0&0\\
    f''(q)&0&0&0&0\\
    0&0&-\hat\beta_1f''(q^{11}_0)-r^{11}_0f'''(q^{11}_0)&-f''(q_0^{11})&0\\
    0&0&-f''(q_0^{11})&0&0\\
    0&0&0&0&0
  \end{bmatrix}
\end{equation}
and where the matrix $C$ encodes the coefficients of the quadratic form
\eqref{eq:inverse.quadratic.form}, and is given element-wise by
\begin{align}
  \notag
  &
  C_{11}=d^{00}_\mathrm df'(1)
  \qquad
  C_{12}=r^{00}_\mathrm df'(1)
  \qquad
  C_{22}=-f'(1)
  \\
  \notag
  &
  C_{13}
  =\frac1{1-q_0}\left(
    (r^{11}_d-r^{11}_0)\left(r^{01}-q\frac{r^{11}_d-r^{11}_0}{1-q_0}\right)(f'(1)-f'(q_0))+qf'(1)d^{00}_d+r^{00}_d(r^{10}f'(1)+(r^{11}_d-r^{11}_0)f'(q))
    \right)
  \\
  \notag
  &
  C_{15}=r^{00}_df'(q)+\left(r^{01}-q\frac{r^{11}_d-r^{11}_0}{1-q_0}\right)(f'(1)-f'(q_0))
  \qquad
  C_{14}=-C_{15}
  \\
  &
  C_{23}=\frac1{1-q_0}\left((qr^{00}_d-r^{10})f'(1)-(r^{11}_d-r^{11}_0)f'(q)\right)
  \qquad
  C_{24}=f'(q)
  \qquad
  C_{25}=-C_{24}
  \\
  \notag
  &
  C_{33}
  =
    -\frac{r^{11}_d-r^{11}_0}{1-q_0}\left[
      \frac{r^{11}_d-r^{11}_0}{1-q_0}f'(1)
      -2\left(
        \frac{qr^{01}-r^{11}_0}{1-q_0}+\frac{1-q^2}{1-q_0}\frac{r^{11}_d-r^{11}_0}{1-q_0}
        \right)(f'(1)-f'(q_0))
        -2\frac{qr^{00}-r^{10}}{1-q_0}f'(q)
    \right]\\
  \notag
  &\qquad-\frac{1-q^2}{(1-q_0)^2}-\frac{(r^{10}-qr^{00}_d)^2}{(1-q_0)^2}f'(1)
    \\
  \notag
  &
  C_{34}
  =-(qr^{01}-r^{11}_0)\frac{f'(1)-f'(q_0)}{1-q_0}-\frac{r^{11}_d-r^{11}_0}{1-q_0}\left(
    \frac{1-q^2}{1-q_0}(f'(1)-f'(q_0))-f'(q_0)
    \right)-f'(q)\frac{qr^{00}_d-r^{10}}{1-q_0}
    \\
  \notag
  &
    C_{35}=-C_{34}-\frac{r^{11}_d-r^{11}_0}{1-q_0}(f'(1)-f'(q_0))
  \qquad
    C_{44}=f'(1)-2f'(q_0)
    \qquad
    C_{45}=f'(q_0)
    \qquad
    C_{55}=-f'(1)
  \notag
\end{align}
The saddle point conditions read
\begin{align}
  0=-\beta^2f''(1)a_0+\frac{a_0-\mathcal X^TC\mathcal X}{(1-a_0)^2}
  &&
  0=\bigg(\beta B-\frac1{1-a_0}C\bigg)\mathcal X
\end{align}
Note that the second of these conditions implies that the quadratic form in
$\mathcal X$ in the action vanishes at the saddle.

We would like to take the limit of $\beta\to\infty$. As is usual in the
two-spin model, the appropriate limit of the order parameter is
$a_0=1-(y\beta)^{-1}$. Upon inserting this scaling and taking the limit, we
finally find
\begin{equation}
  \lambda_\mathrm{min}=-2\lim_{\beta\to\infty}\frac1\beta\mathcal S_x
  =\mu-\left(y+\frac1yf''(1)\right)
\end{equation}
with associated saddle point conditions
\begin{align}
  0=-f''(1)+y^2(1-\mathcal X^TC\mathcal X)
  &&
  0=(B-yC)\mathcal X
\end{align}
The trivial solution, which gives the bottom of the semicircle, is for
$\mathcal X=0$. When this is satisfied, the first equation gives $y^2=f''(1)$,
and
\begin{equation}
  \lambda_\mathrm{min}=\mu-\sqrt{4f''(1)}=\mu-\mu_\mathrm m
\end{equation}
as expected. The nontrivial solutions have nonzero $\mathcal X$. The only way
to satisfy this with the saddle conditions is for $y$ such that one of the
eigenvalues of $B-yC$ is zero. In this case, if the normalized eigenvector
associated with the zero eigenvector is $\hat{\mathcal X}_0$, $\mathcal
X=\|\mathcal X_0\|\hat{\mathcal X}_0$ is a solution. The magnitude of the solution
is set by the other saddle point condition, namely
\begin{equation}
  \|\mathcal X_0\|^2=\frac1{\hat{\mathcal X}_0^TC\hat{\mathcal X}_0}\left(1-\frac{f''(1)}{y^2}\right)
\end{equation}
In practice, we find that $\hat{\mathcal X}_0^TC\hat{\mathcal X}_0$ is positive
at the saddle point. Therefore, for the solution to make sense we must have
$y^2\geq f''(1)$. In practice, there is at most \emph{one} $y$ which produces a
zero eigenvalue of $B-yC$ and satisfies this inequality, so the solution seems
to be unique.

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}
  q_\textrm{min}=\frac{\mathbf x_{0\leftarrow 1}\cdot\mathbf x_\mathrm{min}}N
  =\frac{x_0}{1-q}
\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}

\section{Franz--Parisi potential}
\label{sec:franz-parisi}

Here, we compute the Franz--Parisi potential for this model 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
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
\begin{equation} \label{eq:franz-parisi.definition}
  \beta V_\beta(q\mid E_0,\mu_0)
  =-\frac1N\overline{\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)}\,
  \log\bigg(\int d\mathbf s\,\delta\big(\|\mathbf s\|^2-N\big)\,\delta(\pmb\sigma\cdot\mathbf s-Nq)\,e^{-\beta H(\mathbf s)}\bigg)}
\end{equation}
Both the denominator and the logarithm are treated using the replica trick, which yields
\begin{equation}
  \beta V_\beta(q\mid E_0,\mu_0)
  =-\frac1N\lim_{\substack{m\to0\\n\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\mathbf s_a\,\delta(\|\mathbf s_a\|^2-N)\,\delta(\pmb \sigma_1\cdot \mathbf s_a-Nq)\,e^{-\beta H(\mathbf s_a)}\right)}
\end{equation}
The derivation of this proceeds in much the same way as for the complexity or
the isolated eigenvalue. Once the $\delta$-functions are converted to
exponentials, the $H$-dependent terms can be expressed by convolution with the
linear operator
\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)
    -\beta
    \sum_a^n\delta(\mathbf t-\mathbf s_a)
\end{equation}
Averaging over $H$ squares the application of this operator to $f$ as before.
After performing a Hubbard--Stratonovich using matrix order parameters
identical to those used in the calculation of the complexity, we find that
\begin{equation}
  \beta V_\beta(q\mid E_0,\mu_0)=-\frac1N\lim_{\substack{m\to0\\n\to0}}\frac\partial{\partial n}\int d\mathcal Q_0\,d\mathcal Q_1\,e^{Nm\mathcal S_0(\mathcal Q_0)+Nn\mathcal S_\mathrm{FP}(\mathcal Q_1\mid\mathcal Q_0)}
\end{equation}
where $\mathcal S_0$ is the same as in \eqref{eq:one-point.action} and
\begin{equation}
  n\mathcal S_{\mathrm{FP}}
  =\frac12\beta^2\sum_{ab}^nf(Q_{ab})
  +\beta\sum_a^m\sum_b^n\left[
    \hat\beta_0f(C^{01}_{ab})
    +R^{10}_{ab}f'(C^{01}_{ab})
  \right]
  +\frac12\log\det\left(
    Q-\begin{bmatrix}C^{01}\\iR^{10}\end{bmatrix}^T\begin{bmatrix}C^{00}&iR^{00}\\iR^{00}&D^{00}\end{bmatrix}^{-1}\begin{bmatrix}C^{01}\\iR^{10}\end{bmatrix}
    \right)
\end{equation}
Here, because we are at low but nonzero temperature for the constrained configuration,
we make a {\oldstylenums1}\textsc{rsb} anstaz for the matrix $Q$, while the
$00$ matrices will take their saddle point value for the one-point complexity
and the $01$ matrices have the same structure as \eqref{eq:01.ansatz}.
Inserting these gives
\begin{equation}
  \begin{aligned}
    \beta V_\beta&=\frac12\beta^2\big[f(1)-(1-x)f(q_1)-xf(q_0)\big]
    +\beta\hat\beta_0f(q)+\beta r^{10}f'(q)-\frac{1-x}x\log(1-q_1)
    \\
                 &\qquad+\frac1x\log(1-(1-x)q_1-xq_0)
     +\frac{q_0-d^{00}_df'(1)q^2-2r^{00}_df'(1)r^{10}q+(r^{10})^2f'(1)}{
      1-(1-x)q_1-xq_0
    }
  \end{aligned}
\end{equation}
The saddle point for $r^{10}$ can be taken explicitly. After this, we take the
limit of $\beta\to\infty$. There are two possibilities. First, in the replica
symmetric case $x=1$, and in the limit of large $\beta$ $q_0$ will scale like
$q_0=1-(y_0\beta)^{-1}$. Inserting this, the limit is
\begin{equation}
  V_\infty^{\textsc{rs}}=-\hat\beta_0 f(q)-r^{00}_\mathrm df'(q)q-\frac12\left(y_0(1-q^2)+\frac{f'(1)^2-f'(q)^2}{y_0f'(1)}\right)
\end{equation}
The saddle point in $y_0$ can now be taken, taking care to choose the solution for $y_0>0$. This gives
\begin{equation}
  V_\infty^{\textsc{rs}}(q\mid E_0,\mu_0)=-\hat\beta_0f(q)-r^{00}_\mathrm df'(q)q-\sqrt{(1-q^2)\left(1-\frac{f'(q)^2}{f'(1)^2}\right)}
\end{equation}
The second case is when the inner statistical mechanics problem has replica
symmetry breaking. Here, $q_0$ approaches a nontrivial limit, but
$x=z\beta^{-1}$ approaches zero and $q_1=1-(y_1\beta)^{-1}$ approaches one. The result is
\begin{equation}
  \begin{aligned}
    V_\infty^{\oldstylenums{1}\textsc{rsb}}(q\mid E_0,\mu_0)
    &=-\hat\beta_0f(q)-r^{00}_\mathrm df'(q)q-\frac12\bigg(
      z(f(1)-f(q_0))+\frac{f'(1)}{y_1}-\frac{y_1(q^2-q_0)}{1+y_1z(1-q_0)} \\
    &\hspace{8pc}-(1+y_1z(1-q_0))\frac{f'(q)^2}{y_1f'(1)}+\frac1z\log\left(1+zy_1(1-q_0)\right)
    \bigg)
  \end{aligned}
\end{equation}
Though the saddle point in $y_1$ can be evaluated in this expression, it
delivers no insight. The final potential is found by taking the saddle over
$z$, $y_1$, and $q_0$. A plot comparing the result to the minimum energy
saddles is found in Fig.~\ref{fig:franz-parisi}. As noted above, there is
little qualitatively different from what was found in \cite{Ros_2019_Complexity}
for the pure models.

\section{Conclusion}
\label{sec:conclusion}

We have computed the complexity of neighboring stationary points for the mixed
spherical models. When we studied the neighborhoods of marginal minima, we
found something striking: only those at the threshold energy have other
marginal minima nearby. For the many marginal minima away from the threshold
(including the exponential majority), there is a gap in overlap between them.

This has implications for pictures of relaxation and aging. In most $p+s$
models studied, quenches from infinite to zero temperature (gradient descent
starting from a random point) relax towards marginal states with energies above
the threshold energy \cite{Folena_2023_On}, while at least in some models a
quench to zero temperature from a temperature around the dynamic transition
relaxes towards marginal states with energies below the threshold energy
\cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially
Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the
neighborhoods of marginal states above and below the threshold are quite
different, and yet the emergent aging behaviors relaxing toward states above and
below the threshold seem to be the same. Therefore, this kind of dynamics
appears to be insensitive to the neighborhood of the marginal state being
approached. To understand something better about why certain states attract the
dynamics in certain situations, nonlocal information, like the
structure of their entire basin of attraction, seems vital.

It is possible that replica symmetry breaking among the constrained stationary
points could change the details of the two-point complexity of very nearby
states. Indeed, it is difficult to rule out \textsc{rsb} in complexity
calculations. However, such corrections would not change the overarching
conclusions of this paper, namely that most marginal minima are separated from
each other by a macroscopic overlap gap and high barriers. This is because the
replica symmetric complexity bounds any \textsc{rsb} complexities from above,
and so \textsc{rsb} corrections can only decrease the complexity. Therefore,
the overlap gaps, which correspond to regions of negative complexity, cannot be
removed by a more detailed saddle point ansatz.

Our calculation studied the neighborhood of typical reference points with the
given energy and stability. However, it is possible that marginal minima with
atypical neighborhoods actually attract the dynamics. To determine this, a
different type of calculation is needed. As our calculation is akin to the
quenched Franz--Parisi potential, study of atypical neighborhoods would entail
something like the annealed Franz--Parisi approach, i.e.,
\begin{equation}
  \Sigma^*(E_0,\mu_0,E_1,\mu_1,q)=\frac1N\overline{\log\left(
      \int d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)\,d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)
  \right)}
\end{equation}
which puts the two points on equal footing. This calculation and exploration of
the atypical neighborhoods it reveals is a clear future direction.

The methods developed in this paper are straightforwardly (if not easily)
generalized to landscapes with replica symmetry broken complexities
\cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features
of this study would persist, with neighboring states being divided into
different clusters based on the \textsc{rsb} order but with the basic presence
or absence of overlap gaps and the nature of the stability of near-neighbors
remaining unchanged. Interesting structure might emerge in the arrangement of
marginal states in \textsc{frsb} systems, where the ground state itself is
marginal and coincides with the threshold.

\paragraph{Acknowledgements}

The author would like to thank Valentina Ros, Giampaolo Folena, Chiara
Cammarota, and Jorge Kurchan for useful discussions related to this work.

\paragraph{Funding information}

JK-D is supported by a \textsc{DynSysMath} Specific Initiative by the
INFN.

\printbibliography

\end{document}