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\title{
  When is the average number of saddle points typical?
}

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

\abstract{
  A common measure of a function's complexity is the count of its stationary
  points. For complicated functions, this count grows exponentially with the
  volume and dimension of their domain. In practice, the count is averaged over
  a class of functions (the annealed average), but the large numbers
  involved can produce averages biased by extremely rare samples. Typical
  counts are reliably found by taking the average of the logarithm (the
  quenched average), which is more difficult and not often done in practice.
  When most stationary points are uncorrelated with each other, quenched and
  annealed averages are equal. Equilibrium heuristics can guarantee when most of
  the lowest minima will be uncorrelated. We show that these equilibrium
  heuristics cannot be used to draw conclusions about other minima and saddles
  by producing examples among Gaussian-correlated functions on the hypersphere
  where the count of certain saddles and minima has different quenched and
  annealed averages, despite being guaranteed `safe' in the equilibrium
  setting. We determine conditions for the emergence of nontrivial correlations
  between saddles, and discuss the implications for the geometry of those
  functions and what out-of-equilibrium settings might be affected.
}

\begin{document}

\maketitle

Random high-dimensional energies, cost functions, and interaction networks are
important across disciplines: the energy landscape of glasses, the likelihood
landscape of machine learning and inference, and the interactions between
organisms in an ecosystem are just a few examples \cite{Stein_1995_Broken,
Krzakala_2007_Landscape, Altieri_2021_Properties, Yang_2023_Stochastic}. A
traditional tool for making sense of their behavior is to analyze the
statistics of points where their dynamics are stationary
\cite{Cavagna_1998_Stationary, Fyodorov_2004_Complexity, Fyodorov_2007_Density,
Bray_2007_Statistics}. For energy or cost landscapes, these correspond to the
minima, maxima, and saddles, while for ecosystems and other non-gradient
dynamical systems these correspond to equilibria of the dynamics. When many
stationary points are present, the system is considered complex.

Despite the importance of stationary point statistics for understanding complex
behavior, they are often calculated using an uncontrolled approximation.
Because their number is so large, it cannot be reliably averaged. The annealed
approximation takes this average anyway, risking a systematic bias by rare and
atypical samples. The annealed approximation is known to be exact for certain
models and in certain circumstances, but it is used outside those circumstances
without much reflection \cite{Wainrib_2013_Topological,
  Kent-Dobias_2021_Complex, Gershenzon_2023_On-Site}. In a few cases
  researchers have instead made the better-controlled quenched average, which
  averages the logarithm of the number of stationary points, and find
  deviations from the annealed approximation with important implications for
  behavior \cite{Cavagna_1999_Quenched, Crisanti_2006_Spherical,
  Muller_2006_Marginal, Ros_2019_Complex, Kent-Dobias_2023_How,
Ros_2023_Quenched, Ros_2023_Generalized}. Generically, the annealed
approximation to the complexity is wrong when a nonvanishing fraction of pairs
of stationary points have nontrivial correlations in their mutual position.

A heuristic line of reasoning for the appropriateness of the annealed
approximation is sometimes made when the approximation is correct for an
equilibrium calculation on the same system. The argument goes like this: since
the limit of zero temperature in an equilibrium calculation concentrates the
Boltzmann measure onto the lowest set of minima, the equilibrium free energy in
the limit to zero temperature will be governed by the same statistics as the
count of that lowest set of minima. This argument is strictly valid only for
the lowest minima, which at least in glassy problems are rarely relevant to
dynamical behavior. What about the \emph{rest} of the stationary points?

In this paper, we show that the behavior of the ground state, or \emph{any}
equilibrium behavior, does not govern whether stationary points will have a
correct annealed average. In a prototypical family of models of random
functions, we determine a condition for when annealed averages should fail and
some stationary points will have nontrivial correlations in their mutual
position. We produce examples of models whose equilibrium is guaranteed to
never see such correlations between thermodynamic states, but where a
population of saddle points is nevertheless correlated.

We study the mixed spherical models, which are models of Gaussian-correlated
random functions with isotropic statistics on the $(N-1)$-sphere. Each model
consists of a class of functions $H:S^{N-1}\to\mathbb R$ defined by the
covariance between the functions evaluated at two different points
$\pmb\sigma_1,\pmb\sigma_2\in S^{N-1}$, which is a function of the scalar
product (or overlap) between the two configurations:
\begin{equation} \label{eq:covariance}
  \overline{H(\pmb\sigma_1)H(\pmb\sigma_2)}=\frac1Nf\bigg(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\bigg)
\end{equation}
We will further take the distribution of $H$ to be centered, i.e., $\overline{H(\pmb\sigma)}=0$ for all $\pmb\sigma\in S^{N-1}$, which is equivalent to the absence of any deterministic term (or spike) in the function.
Specifying the covariance function $f$ uniquely specifies the model. The series
coefficients of $f$ need to be nonnnegative in order for $f$ to be a
well-defined covariance. The case where $f$ is a homogeneous polynomial has
been extensively studied, and corresponds to the pure spherical models of glass
physics or the spiked tensor models of statistical inference
\cite{Castellani_2005_Spin-glass}. Here our examples will be models with
$f(q)=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$ for $\lambda\in(0,1)$,
called $3+s$ models.\footnote{
  Though the examples and discussion will focus on the $3+s$ models, most
  formulas (including the principal result in \eqref{eq:condition}) are valid for
  arbitrary covariance functions $f$ under the condition that $f'(0)=0$, i.e.,
  that there is no linear field in the problem. This condition is necessary to
  ensure that what we call `trivial' correlations are actually \emph{zero}
  correlations: in the absence of a field, trivially correlated points on the
  sphere are orthogonal. This simplifies our formulas by setting the overlap
  $q_0$ between trivially correlated configurations to zero, which would
  otherwise be another order parameter, but reduces the scope of this study. The
  trivial overlap $q_0$ is also important in situations where a deterministic
  field (or spike) is present, as in \cite{Ros_2019_Complex}, but deterministic
  fields are likewise not considered here.
} These are examples of \emph{mixed} spherical models, which have been studied
in the physics and statistics literature and host a zoo of complex orders and
phase transitions \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
  Krakoviack_2007_Comment, Crisanti_2007_Amorphous-amorphous,
Crisanti_2011_Statistical, BenArous_2019_Geometry, Subag_2020_Following,
ElAlaoui_2020_Algorithmic}.


There are several well-established results on the equilibrium of this model.
First, if the function $\chi(q)=f''(q)^{-1/2}$ is convex then it is not possible for the
equilibrium solution to have nontrivial correlations between states at any
temperature \cite{Crisanti_2006_Spherical, Crisanti_2007_Amorphous-amorphous}.\footnote{
  More specifically, convex $\chi$ cannot have an equilibrium order with more than
  {\oldstylenums1\textsc{rsb}} order among the configurations. In equilibrium,
  {\oldstylenums1\textsc{rsb}} corresponds to trivial correlations between
  thermodynamic states, but nontrivial correlations exist \emph{within} a state
  at nonzero temperature. When temperature goes to zero,
  {\oldstylenums1\textsc{rsb}} in equilibrium reduces to replica symmetry among
  the lowest-lying states. Because in this paper we focus on symmetry breaking
  between stationary points, we consider this form of \textsc{rsb} in
  equilibrium trivial because it does not imply any nontrivial correlations
  between states.
}
This is a strong condition on the form of equilibrium order. Note that
non-convex $\chi$ does not imply that you \emph{will} see nontrivial correlations between
states at some temperature. In the $3+s$ models we consider here, models with
$s>8$ have non-convex $\chi$ and those with $s\leq8$ have convex $\chi$ independent
of $\lambda$. Second, the characterization of the ground state has been made
\cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
Crisanti_2011_Statistical, Auffinger_2022_The}. In the $3+s$ models we
consider, for $s>12.430...$ nontrivial ground state configurations (more than {\oldstylenums1\textsc{rsb}}) appear in
a range of $\lambda$. These bounds on equilibrium order are shown in
Fig.~\ref{fig:phases}, along with our result for where the complexity has
nontrivial correlations between some stationary points. As evidenced in that
figure, correlations among saddles are possible well inside regions that
forbid them among equilibrium states.

There are two important features which differentiate stationary points
$\pmb\sigma^*$ in the spherical models: their \emph{energy density}
$E=\frac1NH(\pmb\sigma^*)$ and their \emph{stability}
$\mu=\frac1N\operatorname{\mathrm{Tr}}\operatorname{\mathrm{Hess}}H(\pmb\sigma^*)$.
The energy density gives the `height' in the landscape, while the
stability governs the spectrum of the stationary point.
In each spherical model, the spectrum of every stationary point is a Wigner
semicircle of the same width $\mu_\mathrm m=\sqrt{4f''(1)}$, but shifted by
constant. The stability $\mu$ sets this constant shift. When $\mu<\mu_\mathrm
m$, the spectrum has support over zero and we have saddles with an
extensive number of downward directions. When $\mu>\mu_\mathrm m$ the spectrum
has support only over positive eigenvalues, and we have stable minima.\footnote{
  Saddle points with a subextensive number of downward directions also exist
  via large deviations of some number of eigenvalues from the average spectrum.
} When $\mu=\mu_\mathrm m$, the spectrum has a pseudogap, and we have marginal minima.

\begin{figure}
  \centering
  \includegraphics[width=0.95\columnwidth]{figs/phases_34.pdf}
  \caption{
    A phase diagram of the boundaries we discuss in this paper for the $3+s$
    model with $f=\frac12\big(\lambda q^3+(1-\lambda)q^s\big)$. The blue region
    shows models which have some stationary points with nontrivial correlated
    (\textsc{rsb}) structure, and is given by $G_f>0$ where $G_f$ is found in
    \eqref{eq:condition}. The yellow region shows where $\chi(q)=f''(q)^{-1/2}$
    is not convex and therefore nontrivial correlations between states are
    possible in equilibrium. The green region shows where nontrivial
    correlations exist at the ground state, adapted from
    \cite{Auffinger_2022_The}. We find that models where correlations between
    equilibrium states are forbidden can nonetheless harbor
    correlated stationary points.
  } \label{fig:phases}
\end{figure}

The number $\mathcal N(E,\mu)$ of stationary points with energy density $E$ and
stability $\mu$ is exponential in $N$. Their complexity
$\Sigma(E,\mu)$ is defined by the average of the logarithm of their number:
$\Sigma(E,\mu)=\frac1N\overline{\log\mathcal N(E,\mu)}$. More often the
annealed complexity is calculated, where the average is taken before the
logarithm: $\Sigma_\mathrm a(E,\mu)=\frac1N\log\overline{\mathcal N(E,\mu)}$.
The annealed complexity has been computed for these models
\cite{BenArous_2019_Geometry, Folena_2020_Rethinking}, and the quenched
complexity has been computed for a couple examples which have nontrivial ground
states \cite{Crisanti_2006_Spherical ,Kent-Dobias_2023_How}. The annealed complexity bounds the
complexity from above. A positive complexity indicates the presence of an
exponentially large number of stationary points of the indicated kind, while a
negative one means it is vanishingly likely they will appear. The line of
zero complexity is significant as the transition between many stationary points
and none.

In these models, trivial correlations between stationary points correspond with
zero overlap: almost all stationary points are orthogonal to each other. This
corresponds with \emph{replica symmetric} (\textsc{rs}) order. The emergence of
nontrivial correlations, and the invalidity of the annealed approximation, occurs
when some non-vanishing fraction of stationary point pairs have a nonzero
overlap. This corresponds to some kind of \emph{replica symmetry breaking}
(\textsc{rsb}). Here we restrict ourselves to a {\oldstylenums1}\textsc{rsb}
ansatz, which corresponds to two kinds of pairs of stationary point: a fraction
$x$ of pairs have the trivial zero overlap, and the remaining fraction $1-x$
have a nontrivial overlap $q_1$. In the annealed or replica-symmetric case,
$x=1$ and all but a vanishing fraction of stationary points are uncorrelated
with each other. Since other kinds of \textsc{rsb} order encompass
{\oldstylenums1}\textsc{rsb}, we are guaranteed that
$\Sigma\leq\Sigma_{\oldstylenums1\textsc{rsb}}\leq\Sigma_\mathrm a$. We will
discuss later in what settings the {\oldstylenums1}\textsc{rsb} complexity is
correct.

When the complexity is calculated using the Kac--Rice formula and a physicists'
tool set, the problem is reduced to the evaluation of an integral by the saddle
point method for large $N$ \cite{Kent-Dobias_2023_How}. The complexity is given
by extremizing an effective action,
\begin{align}
  &\Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu) \notag \\
  &\quad=\lim_{n\to0}\int dq_1\,dx\,\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)e^{nN\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)} \notag \\
  &\quad=\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
\end{align}
for the action $\mathcal S_{\oldstylenums1\textsc{rsb}}$ given by \eqref{eq:1rsb.action}.
\begin{widetext}
  \begin{equation} \label{eq:1rsb.action}
  \begin{aligned}
    &\mathcal S_{\oldstylenums1\textsc{rsb}}(q_1,x\mid E,\mu)
    =\mathcal D(\mu)
    +\mathop{\textrm{extremum}}_{\hat\beta,r_\mathrm d,r_1,d_\mathrm d,d_1}
    \Bigg\{
      \hat\beta E-r_\mathrm d\mu\\
      &\quad+\frac12\bigg[
        \hat\beta^2\big[f(1)-\Delta xf(q_1)\big]
        +(2\hat\beta r_\mathrm d-d_\mathrm d)f'(1)
        -\Delta x(2\hat\beta r_1-d_1)f'(q_1)
        +r_\mathrm d^2f''(1)-\Delta x\,r_1^2f''(q_1) \\
        &\quad+\frac1x\log\Big(
          \big(r_\mathrm d-\Delta x\,r_1\big)^2+d_\mathrm d\big(1-\Delta x\,q_1\big)-\Delta x\,d_1\big(1-\Delta xq_1\big)
          \Big)
          -\frac{\Delta x}x\log\Big(
            (r_\mathrm d-r_1)^2+(d_\mathrm d-d_1)(1-q_1)
          \Big)
      \bigg]
    \Bigg\}
  \end{aligned}
\end{equation}
where $\Delta x=1-x$ and
\begin{equation} \label{eq:hess.term}
  \mathcal D(\mu)
  =\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{equation}
\end{widetext}
The details of the derivation of these expressions can be found in
\cite{Kent-Dobias_2023_How}. The extremal problem in $\hat\beta$, $r_\mathrm
d$, $r_1$, $d_\mathrm d$, and $d_1$ has a unique solution and can be found
explicitly, but the resulting formula is unwieldy. The action can have multiple
extrema, but the one for which the complexity is \emph{smallest} gives the
correct solution. There is always a solution for $x=1$ which is independent of
$q_1$, corresponding to the replica symmetric case, and with $\Sigma_\mathrm
a(E,\mu)=\mathcal S_{\oldstylenums1\textsc{rsb}}(E,\mu\mid q_1,1)$. The crux of
this paper will be to determine when this solution is not the global one.

It isn't accurate to say that a solution to the saddle point equations is
`stable' or `unstable.' The problem of solving the complexity in this way is
not a variational problem, so there is nothing to be maximized or minimized,
and in general even global solutions are not even local minima of the action.
However, the stability of the action can still tell us something about the
emergence of new solutions: when a new solution bifurcates from an existing
one, the action will have a flat direction. Unfortunately this is difficult to
search out, since one must know the parameters of the new solution, and $q_1$
is unconstrained and can take any value in the old solution.

There is one place where we can consistently search for a bifurcating solution
to the saddle point equations: along the zero complexity line $\Sigma_\mathrm
a(E,\mu)=0$. Going along this line in the replica symmetric solution, the
{\oldstylenums1}\textsc{rsb} complexity transitions at a critical point where
$x=q_1=1$ \cite{Kent-Dobias_2023_How}. Since all the parameters in the
bifurcating solution are known at this point, we can search for it by looking
for a flat direction. In the annealed solution for points describing saddles
(with $\mu^2\leq\mu_\mathrm m^2$ and therefore the simpler form of
\eqref{eq:hess.term}), this line is
\begin{equation} \label{eq:extremal.line}
  \mu_0=-\frac{2Ef'f''}{z_f}-\sqrt{\frac{2f''u_f}{z_f^2}\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}
\end{equation}
where we have chosen the lower branch as a convention (see
Fig.~\ref{fig:complexity_35}) and where we define for brevity (here and
elsewhere) the constants
\begin{align}
  u_f&=f(f'+f'')-f'^2
  &&
  v_f=f'(f''+f''')-f''^2 \notag \\
  w_f&=2f''(f''-f')+f'f'''
  &&
  y_f=f'(f'-f)+f''f \\
  z_f&=f(f''-f')+f'^2 \notag
\end{align}
When $f$ and its derivatives appear without an argument, the implied argument
is always 1, so, e.g., $f'\equiv f'(1)$. If $f$ has at least two nonzero
coefficients at second order or higher, all of these constants are positive.
Though in figures we focus on the lower branch of saddles, another set of
identical solutions always exists for $(E,\mu)\mapsto(-E,-\mu)$. We also define
$E_\textrm{min}$, the minimum energy at which saddle points with an extensive
number of downward directions are found, as the energy for which
$\mu_0(E_\mathrm{min})=\mu_\mathrm m$.

Let $M$ be the matrix of double partial derivatives of the action with
respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point
$x=1$ with the additional constraint that $q_1=1$ and along the extremal
complexity line \eqref{eq:extremal.line}. We determine when a zero eigenvalue
appears, indicating the presence of a bifurcating {\oldstylenums1}\textsc{rsb}
solution, by solving $0=\det M$. We find
\begin{equation}
  \det M
  =-\bigg(\frac{\partial^2\mathcal S_{\oldstylenums1\textsc{rsb}}}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\bigg)^2
  \propto(ay^2+bE^2+2cyE-d)^2
\end{equation}
where $y=-\frac12z_f\mu-f'f''E$ is proportional to the square-root term in
\eqref{eq:extremal.line} and the constants $a$, $b$, $c$, and $d$ are defined
by
\begin{equation}
  \begin{aligned}
    a&=\frac{w_f\big(3y_f^2-4ff'f''(f'-f)\big)-6y_f^2(f''-f')f''}{(u_fz_ff'')^2f'}
    \\
    b&=\frac{f'w_f}{z_f^2}
    \qquad
    c=\frac{w_f}{f''z_f^2}
    \qquad
    d=\frac{w_f}{f'f''}
  \end{aligned}
\end{equation}
Changing variables from $\mu$ to $y$ is convenient because the branch
of \eqref{eq:extremal.line} is chosen by the sign of $y$ (the lower-energy
branch we are interested in corresponds with $y>0$). The relationship
between $y$ and $E$ on the extremal line is $g=2hy^2+eE^2$, where the constants
$e$, $g$, and $h$ are given by
\begin{equation}
  e=f''-f'
  \qquad
  g=z_f\log\frac{f''}{f'}
  \qquad
  h=\frac1{f''u_f}
\end{equation}

\begin{figure}
  \centering
  \includegraphics[width=0.95\columnwidth]{figs/complexity_35.pdf}

  \caption{
    Stationary point statistics as a function of energy density $E$ and
    stability $\mu$ for a model with $f(q)=\frac12(\frac12q^3+\frac12q^5)$. The
    dashed black line shows the line of zero annealed complexity and enclosed
    inside the annealed complexity is positive. The solid black line (only
    visible in the inset) gives the line of zero {\oldstylenums1\textsc{rsb}}
    complexity. The red region (blown up in the inset) shows where the annealed
    complexity gives the wrong count and a {\oldstylenums1}\textsc{rsb}
    complexity in necessary. The red points show where $\det M=0$. The left
    point, which is only an upper bound on the transition, coincides with it in
    this case. The gray shaded region highlights the minima, which are
    stationary points with $\mu\geq\mu_\mathrm m$. $E_\textrm{min}$ is marked
    on the plot as the lowest energy at which saddles of extensive index are
    found.
  } \label{fig:complexity_35}
\end{figure}

The solutions for $\det M=0$ can be calculated explicitly and correspond to
energies that satisfy
\begin{equation} \label{eq:energies}
  E_{\oldstylenums1\textsc{rsb}}^\pm
  =\frac{\operatorname{sign}(bg-de)\big(-cg\pm\sqrt{\Delta_f}\big)}
  {
    \sqrt{2c^2eg+(2bh-ae)(bg-de)\mp2ce\sqrt{\Delta_f}}
  }
\end{equation}
where the discriminant $\Delta_f$ is given by
\begin{equation}
  \Delta_f=c^2g^2+(2dh-ag)(bg-de)
\end{equation}
This predicts two points where a {\oldstylenums1}\textsc{rsb} solution can
bifurcate from the annealed one. The remainder of the transition line can be
found by solving the extremal problem for the action very close to one
of these solutions, and then taking small steps in the parameters $E$ and $\mu$
until it terminates. In many cases considered here, the line of transitions in
the complexity that begins at $E_{\oldstylenums1\textsc{rsb}}^+$, the higher
energy point, ends exactly at $E_{\oldstylenums1\textsc{rsb}}^-$, the lower
energy point, so that these two points give the precise range of energies at
which \textsc{rsb} saddles are found. An example that conforms with this
picture for a $3+5$ mixed model is shown in Fig.~\ref{fig:complexity_35}. In
that figure, the range of $\mu$ with {\oldstylenums1\textsc{rsb}} ordering at
any fixed $E$ is extremely small. With increasing $s$ the range also increases
(see the example of the $3+16$ model in \cite{Kent-Dobias_2023_How}), but we do
not have any intuition for why this is.

The discriminant $\Delta_f$ inside the square root of \eqref{eq:energies} is
proportional to
\begin{equation} \label{eq:condition}
  \begin{aligned}
    G_f
    &=
    f'\log\frac{f''}{f'}\big[
      3y_f(f''-f')f'''-2(f'-2f)f''w_f
    \big] \\
    &\qquad-2(f''-f')u_fw_f
    -2\log^2\frac{f''}{f'}f'^2f''v_f
  \end{aligned}
\end{equation}
If $G_f>0$, then the bifurcating solutions exist, and there are some saddles
whose complexity is corrected by a {\oldstylenums1\textsc{rsb}} solution.
Therefore, $G_f>0$ is a sufficient condition to see at least
{\oldstylenums1}\textsc{rsb} in the complexity. If $G_f<0$, then there is
nowhere along the extremal line where saddles can be described by such a
complexity, but this does not definitively rule out \textsc{rsb}: the model may
be unstable to different \textsc{rsb} orders, or its phase boundary may simply
not have a critical point on the extremal line. We discuss the former
possibility later in the paper. The range of $3+s$ models where $G_f$ is
positive is shown in Fig.~\ref{fig:phases}.

\begin{figure*}
  \centering
  \includegraphics[width=0.29\textwidth]{figs/range_plot_1.pdf}
  \hspace{-3.4em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_2.pdf}
  \hspace{-3.4em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_3.pdf}
  \hspace{-3.4em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_4.pdf} \\
  \vspace{-2em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_1.pdf}
  \hspace{-3.4em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_2.pdf}
  \hspace{-3.4em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_3.pdf}
  \hspace{-3.4em}
  \includegraphics[width=0.29\textwidth]{figs/range_plot_log_4.pdf}

  \caption{
    The range of energies where \textsc{rsb} saddles are found for the $3+s$
    model with varying $s$ and $\lambda$. In the top row the black line shows
    $E_\textrm{min}$, the minimum energy where saddles are found, and in the
    bottom row this energy is subtracted away to emphasize when the
    \textsc{rsb} region crosses into minima. For most $s$, both the top and
    bottom lines are given by $E_{\oldstylenums1\textsc{rsb}}^\pm$, but for $s=14$
    there is a portion where the low-energy boundary has $q_1<1$. In that plot,
    the continuation of the $E_{\oldstylenums1\textsc{rsb}}^-$ line is shown
    dashed. Also marked is the range of $\lambda$ for which the ground state
    minima are characterized by nontrivial \textsc{rsb}.
  } \label{fig:energy_ranges}
\end{figure*}

Fig.~\ref{fig:energy_ranges} shows the range of energies where nontrivial
correlations are found between stationary points in several $3+s$ models as
$\lambda$ is varied. For models with smaller $s$, such correlations are found
only among saddles, with the boundary never dipping beneath the minimum energy
of saddles $E_\mathrm{min}$. Also, these models have a transition boundary that
smoothly connects $E_{\oldstylenums1\textsc{rsb}}^+$ and
$E_{\oldstylenums1\textsc{rsb}}^-$, so $E_{\oldstylenums1\textsc{rsb}}^-$
corresponds to the lower bound of \textsc{rsb} complexity. For large enough
$s$, the range passes into minima, which is expected as these models have
nontrivial complexity of their ground states. Interestingly, this appears to
happen at precisely the value of $s$ for which nontrivial ground state
configurations appear, $s=12.403\ldots$. This also seems to correspond with the
decoupling of the \textsc{rsb} solutions connected to
$E_{\oldstylenums1\textsc{rsb}}^+$ and $E_{\oldstylenums1\textsc{rsb}}^-$, with
the two phase boundaries no longer corresponding, as in Fig.~\ref{fig:order}.
In these cases, $E_{\oldstylenums1\textsc{rsb}}^-$ sometimes gives the lower
bound, but sometimes it is given by the termination of the phase boundary
extended from $E_{\oldstylenums1\textsc{rsb}}^+$.

\begin{figure}
  \centering
  \includegraphics[width=0.95\columnwidth]{figs/order_plot_1.pdf}\\
  \vspace{-1em}
  \includegraphics[width=0.95\columnwidth]{figs/order_plot_2.pdf}

  \caption{
    Examples of $3+14$ models where
    $E_{\oldstylenums1\textsc{rsb}}^-$ (Top) is the lower bound and (Bottom) is
    not the lower bound of energies with \textsc{rsb} saddles. In
    both plots the red dot shows $E_{\oldstylenums1\textsc{rsb}}^-$, while the
    solid red lines show the transition boundary between \textsc{rs} and
    {\oldstylenums1\textsc{rsb}} complexities. The dashed black line shows the
    \textsc{rs} zero complexity line, while the solid black line shows the
    {\oldstylenums1}\textsc{rsb} zero complexity line. The dashed red lines
    show the spinodals of the {\oldstylenums1\textsc{rsb}} phases. The dotted
    red line shows a discontinuous phase transition between different
    {\oldstylenums1}\textsc{rsb} phases. \textbf{Top:} $\lambda=0.67$. The
    transition line that begins at $E_{\oldstylenums1\textsc{rsb}}^+$ does not
    intersect $E_{\oldstylenums1\textsc{rsb}}^-$ but terminates at a higher
    energy. $E_{\oldstylenums1\textsc{rsb}}^-$ is a lower bound on the energy
    of \textsc{rsb} saddles. There are two competing
    {\oldstylenums1\textsc{rsb}} phases among saddles. \textbf{Bottom:}
    $\lambda=0.69$. The transition line that begins at
    $E_{\oldstylenums1\textsc{rsb}}^+$ terminates at a lower energy than
    $E_{\oldstylenums1\textsc{rsb}}^-$, and therefore its terminus defines the
    lower bound.
  } \label{fig:order}
\end{figure}

There are implications for the emergence of \textsc{rsb} in equilibrium.
Consider a specific $H$ with
\begin{equation}
  \begin{aligned}
    H(\pmb\sigma)
    &=\frac{\sqrt\lambda}{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}\sigma_{i_1}\cdots\sigma_{i_p} \\
    &\hspace{5pc}+\frac{\sqrt{1-\lambda}}{s!}\sum_{i_1\cdots i_s}J^{(s)}_{i_1\cdots i_s}\sigma_{i_1}\cdots\sigma_{i_s}
  \end{aligned}
\end{equation}
where the interaction tensors $J$ are drawn from zero-mean normal distributions
with $\overline{(J^{(p)})^2}=p!/2N^{p-1}$ and likewise for $J^{(s)}$. Functions
$H$ defined this way have the covariance property \eqref{eq:covariance} with
$f(q)=\frac12\big(\lambda q^p+(1-\lambda)q^s\big)$. With the $J$s drawn in this
way and fixed for $p=3$ and $s=14$, we can vary $\lambda$, and according to
Fig.~\ref{fig:phases} we should see a transition in the type of order at the
ground state. What causes the change? Our analysis indicates that stationary
points with the required order \emph{already exist in the landscape} as
unstable saddles for small $\lambda$, then eventually stabilize into metastable
minima and finally become the lowest lying states. This is different from the
picture of existing uncorrelated low-lying states splitting apart into
correlated clusters. Where uncorrelated stationary points do appear to split
apart, when $\lambda$ is decreased from large values, is among saddles, not
minima.

A similar analysis can be made for other mixed models, like the $2+s$, which
should see complexities with other forms of \textsc{rsb}. For instance, in
\cite{Kent-Dobias_2023_How} we show that the complexity transitions from
\textsc{rs} to full \textsc{rsb} (\textsc{frsb}) along the line
\begin{equation}
  \mu
  =-\frac{(f'+f''(0))u_f}{(2f-f')f'f''(0)^{1/2}}
  -\frac{f''-f'}{f'-2f}E
\end{equation}
which can only be realized when $f''(0)\neq0$, as in the $2+s$ models. For
$s>2$, this transition line \emph{always} intersects the extremal line
\eqref{eq:extremal.line}, and so \textsc{rsb} complexity will always be found
among some population of stationary points. However, it is likely that for much
of the parameter space the so-called one-full \textsc{rsb}
({\oldstylenums1\textsc{frsb}}), rather than \textsc{frsb}, is the correct
solution, as it likely is for large $s$ and certain $\lambda$ in the $3+s$
models studied here. Further work to find the conditions for transitions of the
complexity to {\oldstylenums1\textsc{frsb}} and {\oldstylenums2\textsc{frsb}}
is necessary. For values of $s$ where there is trivial \textsc{rsb} in the
ground state, we expect that the {\oldstylenums1\textsc{rsb}} complexity is
correct.

What are the implications for dynamics? We find that nontrivial correlations
tend to exist among saddle points with the largest or smallest possible index
at a given energy density, which are quite atypical in the landscape. However,
these strangely correlated saddle points must descend to uncorrelated minima,
which raises questions about whether structure on the boundary of a basin of
attraction is influential to the dynamics that descends into that basin. These
saddles might act as early-time separatrices for descent trajectories of
certain algorithms.  With open problems in even the gradient decent dynamics on
these models (itself attracted to an atypical subset of marginal minima), it
remains to be seen whether such structures could be influential
\cite{Folena_2020_Rethinking, Folena_2021_Gradient, Folena_2023_On}. This
structure among saddles cannot be the only influence, since it seems that the
$3+4$ model is `safe' from nontrivial \textsc{rsb} among saddles.

We have determined the conditions under which the complexity of the mixed $3+s$
spherical models has different quenched and annealed averages, as the result of
nontrivial correlations between stationary points. We saw that these conditions
can arise among certain populations of saddle points even when the model is
guaranteed to lack such correlations between equilibrium states, and exist for
saddle points at a wide range of energies. This suggests that studies making
complexity calculations cannot reliably use equilibrium behavior to defend the
annealed approximation. Our result has direct implications for the geometry of
these landscapes, and perhaps could be influential to certain
out-of-equilibrium dynamics.

\acknowledgements

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

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\end{document}
