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+\documentclass[reprint,aps,prl,longbibliography,floatfix]{revtex4-2}
+
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{amsmath,amssymb,latexsym,graphicx}
+\usepackage{newtxtext,newtxmath} % Times for PR
+\usepackage[dvipsnames]{xcolor}
+\usepackage[
+  colorlinks=true,
+  urlcolor=MidnightBlue,
+  citecolor=MidnightBlue,
+  filecolor=MidnightBlue,
+  linkcolor=MidnightBlue
+]{hyperref} % ref and cite links with pretty colors
+\usepackage{anyfontsize}
+
+\begin{document}
+
+\title{
+  Unveiling the complexity of hierarchical energy landscapes
+}
+
+\author{Jaron Kent-Dobias}
+\author{Jorge Kurchan}
+\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
+
+\begin{abstract}
+  Complexity is a measure of the number of stationary points in complex
+  landscapes. We derive a general solution for the complexity of mean-field
+  complex landscapes. It incorporates Parisi's solution for the ground state,
+  as it should. Using this solution, we count the stationary points of two
+  models: one with multi-step replica symmetry breaking, and one with full
+  replica symmetry breaking. These examples demonstrate the consistency of the
+  solution and reveal that the signature of replica symmetry breaking at high
+  energy densities is found in high-index saddles, not minima.
+\end{abstract}
+
+\maketitle
+
+The functions used to describe the energies, costs, and fitnesses of disordered
+systems in physics, computer science, and biology are typically \emph{complex},
+meaning that they have a number of minima that grows exponentially with the
+size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex,
+Altieri_2021_Properties}. Though they are often called `rough landscapes' to
+evoke the intuitive image of many minima in something like a mountain range,
+the metaphor to topographical landscapes is strained by the reality that these
+complex landscapes also exist in very high dimensions: think of the dimensions
+of phase space for $N$ particles, or the number of parameters in a neural
+network.
+
+The \emph{complexity} of a function is the average of the logarithm of the
+number of its minima, maxima, and saddle points (collectively stationary
+points), under conditions fixing the value of the energy or the index of the
+stationary point 
+\cite{Bray_1980_Metastable}. 
+Since in complex landscapes this
+number grows exponentially with system size, their complexity is an extensive
+quantity. Understanding the complexity offers an understanding about the
+geometry and topology of the landscape, which can provide insight into
+dynamical behavior.
+
+When complex systems are fully connected, i.e., each degree of freedom
+interacts directly with every other, they are often described by a hierarchical
+structure of the type first proposed by Parisi, the \emph{replica symmetry
+breaking} (RSB) \cite{Parisi_1979_Infinite}. This family of structures is rich, spanning uniform
+\emph{replica symmetry} (RS), an integer $k$ levels of hierarchical nested
+structure ($k$RSB), a full continuum of nested structure (full RSB or FRSB),
+and arbitrary combinations thereof. Though these rich structures are understood
+in the equilibrium properties of fully connected models, the complexity has
+only been computed in RS cases.
+
+In this paper and its longer companion, we share the first results for the
+complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. Using a
+general form for the solution detailed in a companion article, we describe the
+structure of landscapes with a 1RSB complexity and a full RSB complexity
+\footnote{The Thouless--Anderson--Palmer (TAP) complexity is the complexity of
+  a kind of mean-field free energy. Because of some deep thermodynamic
+  relationships between the TAP complexity and the equilibrium free energy, the
+TAP complexity can be computed with extensions of the equilibrium method. As a
+result, the TAP complexity has been previously computed for nontrivial
+hierarchical structure.}.
+
+We study the mixed $p$-spin spherical models, with 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}
+is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
+$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with
+zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
+the energy is typically extensive. The overbar will always denote 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 so-called `pure'
+models have $a_p=1$ for some $p$ and all others zero.
+
+The complexity of the $p$-spin models has been extensively studied by
+physicists and mathematicians. Among physicists, the bulk of work has been on
+ the so-called  `TAP' complexity, 
+which counts minima in the mean-field Thouless--Anderson--Palmer () free energy \cite{Rieger_1992_The,
+Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An,
+Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
+Giardina_2005_Supersymmetry}. The landscape complexity has been proven for pure
+and mixed models without RSB \cite{Auffinger_2012_Random,
+Auffinger_2013_Complexity, BenArous_2019_Geometry}. The mixed models been
+treated without RSB \cite{Folena_2020_Rethinking}. And the methods of
+complexity have been used to study many geometric properties of the pure
+models, from the relative position of stationary points to one another to shape
+and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}.
+
+The variance of the couplings implies that the covariance of the energy with
+itself depends on only the dot product (or overlap) between two configurations.
+In particular, one finds
+\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 $f$ is defined by the series
+\begin{equation}
+  f(q)=\frac12\sum_pa_pq^p.
+\end{equation}
+One needn't start with a Hamiltonian like
+\eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule
+\eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in
+the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.
+
+The family of spherical models thus defined is quite rich, and by varying the
+covariance $f$ nearly any hierarchical structure can be found in
+equilibrium. Because of a correspondence between the ground state complexity
+and the entropy at zero temperature, any hierarchical structure in the
+equilibrium should be reflected in the complexity.
+
+The complexity is calculated using the Kac--Rice formula, which counts the
+stationary points using a $\delta$-function weighted by a Jacobian
+\cite{Kac_1943_On, Rice_1939_The}. The count is given by
+\begin{equation}
+  \begin{aligned}
+    \mathcal N(E, \mu)
+    &=\int_{\mathbb R^N}d\boldsymbol\xi\,e^{-\frac12\|\boldsymbol\xi\|^2/\sigma^2}\int_{S^{N-1}}d\mathbf s\, \delta\big(\nabla H(\mathbf s)-\boldsymbol\xi\big)\,\big|\det\operatorname{Hess}H(\mathbf s)\big| \\
+    &\hspace{2pc}\times\delta\big(NE-H(\mathbf s)\big)\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s)\big)
+  \end{aligned}
+\end{equation}
+with two additional $\delta$-functions inserted to fix the energy density $E$
+and the stability $\mu$. The additional `noise' field $\boldsymbol\xi$
+helps regularize the $\delta$-functions for the energy and stability at finite
+$N$, and will be convenient for defining the order parameter matrices later. The complexity is then
+\begin{equation} \label{eq:complexity}
+  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{\sigma\to0}\frac1N\overline{\log\mathcal N(E, \mu}).
+\end{equation}
+Most of the difficulty of this calculation resides in the logarithm in this
+formula.
+
+The stability $\mu$, sometimes called the radial reaction, determines the depth
+of minima or the index of saddles. At large $N$ the Hessian can be shown to
+consist of the sum of a GOE matrix with variance $f''(1)/N$ shifted by a
+constant diagonal matrix of value $\mu$. Therefore, the spectrum of the Hessian
+is a Wigner semicircle of radius $\mu_\mathrm m=\sqrt{4f''(1)}$ centered at $\mu$. When
+$\mu>\mu_\mathrm m$, stationary points are minima whose sloppiest eigenvalue is
+$\mu-\mu_\mathrm m$. When $\mu=\mu_\mathrm m$, the stationary points are marginal minima with
+flat directions. When $\mu<\mu_\mathrm m$, the stationary points are saddles with
+indexed fixed to within order one (fixed macroscopic index).
+
+It's worth reviewing the complexity for the best-studied case of the pure model
+for $p\geq3$ \cite{Cugliandolo_1993_Analytical}. Here, because the covariance
+is a homogeneous polynomial, $E$ and $\mu$ cannot be fixed separately, and one
+implies the other: $\mu=pE$. Therefore at each energy there is only one kind of
+stationary point. When the energy reaches $E_\mathrm{th}=-\mu_\mathrm m/p$, the
+population of stationary points suddenly shifts from all saddles to all minima,
+and there is an abrupt percolation transition in the topology of
+constant-energy slices of the landscape. This behavior of the complexity can be
+used to explain a rich variety of phenomena in the equilibrium and dynamics of
+the pure models: the `threshold' \cite{Cugliandolo_1993_Analytical} energy $E_\mathrm{th}$ corresponds to the
+average energy at the dynamic transition temperature, and the asymptotic energy
+reached by slow aging dynamics.
+
+Things become much less clear in even the simplest mixed models. For instance,
+one mixed model known to have a replica symmetric complexity was shown to
+nonetheless not have a clear relationship between features of the complexity
+and the asymptotic dynamics \cite{Folena_2020_Rethinking}. There is no longer a
+sharp topological transition.
+
+In the pure models, $E_\mathrm{th}$ also corresponds to the \emph{algorithmic
+threshold} $E_\mathrm{alg}$, defined by the lowest energy reached by local
+algorithms like approximate message passing \cite{ElAlaoui_2020_Algorithmic,
+ElAlaoui_2021_Optimization}. In the spherical models, this has been proven to
+be
+\begin{equation}
+  E_{\mathrm{alg}}=-\int_0^1dq\,\sqrt{f''(q)}
+\end{equation}
+For full RSB systems, $E_\mathrm{alg}=E_0$ and the algorithm can reach the
+ground state energy. For the pure $p$-spin models,
+$E_\mathrm{alg}=E_\mathrm{th}$, where $E_\mathrm{th}$ is the energy at which
+marginal minima are the most common stationary points. Something about the
+topology of the energy function might be relevant to where this algorithmic
+threshold lies.
+
+To compute the complexity in the generic case, we use the replica method to
+treat the logarithm inside the average of \eqref{eq:complexity}, and the
+$\delta$-functions are written in a Fourier basis. The average of the factor
+including the determinant and the factors involving $\delta$-functions can be
+averaged over the disorder separately \cite{Bray_2007_Statistics}. The result
+can be written
+\begin{equation}
+  \Sigma(E,\mu)=\lim_{N\to\infty}\lim_{n\to0}\frac1N\frac{\partial}{\partial n}
+  \int_{\mathrm M_n(\mathbb R)} dQ\,dR\,dD\,e^{N\mathcal S(Q,R,D\mid E,\mu)},
+\end{equation}
+where the effective action $\mathcal S$ is a function of three matrices indexed
+by the $n$ replicas:
+\begin{equation}
+  \begin{aligned}
+    &Q_{ab}=\frac{\mathbf s_a\cdot\mathbf s_b}N
+    \hspace{4em}
+    R_{ab}=\frac{\boldsymbol\xi_a\cdot\mathbf s_b}{N\sigma^2}
+    \\
+    &D_{ab}=\frac1{N\sigma^4}\left(\sigma^2\delta_{ab}-\boldsymbol\xi_a\cdot\boldsymbol\xi_b\right).
+  \end{aligned}
+\end{equation}
+The matrix $Q$ is a clear analogue of the usual overlap matrix of the
+equilibrium case. The matrices $R$ and $D$ have interpretations as response
+functions: $R$ is related to the typical displacement of stationary points by
+perturbations to the potential, and $D$ is related to the change in the
+complexity caused by the same perturbations. The general expression for the
+complexity as a function of these matrices is also found in
+\cite{Folena_2020_Rethinking}.
+
+The complexity is found by the saddle point method, extremizing $\mathcal S$
+with respect to $Q$, $R$, and $D$ and replacing the integral with its integrand
+evaluated at the extremum. We make the \emph{ansatz} that all three matrices have
+a hierarchical structure, and moreover that they share the same hierarchical
+structure. This means that the size of the blocks of equal value of each is the
+same, though the values inside these blocks will vary from matrix to matrix.
+This form can be shown to exactly reproduce the ground state energy predicted
+by the equilibrium solution, a key consistency check.
+
+Along one line in the energy--stability plane the solution takes a simple form:
+the matrices $R$ and $D$ corresponding to responses are diagonal, leaving
+only the overlap matrix $Q$ with nontrivial off-diagonal entries. This
+simplification makes the solution along this line analytically tractable even
+for FRSB. The simplification is related to the presence of an approximate
+supersymmetry in the Kac--Rice formula, studied in the past in the context of
+the TAP free energy \cite{Annibale_2003_Supersymmetric, Annibale_2003_The,
+Annibale_2004_Coexistence}. This line of `supersymmetric' solutions terminates
+at the ground state, and describes the most numerous types of stable minima.
+
+Using this solution, one finds a correspondence between properties of the
+overlap matrix $Q$ at the ground state energy, where the complexity vanishes,
+and the overlap matrix in the equilibrium problem in the limit of zero
+temperature. The saddle point parameters of the two problems are related
+exactly. In the case where the vicinity of the equilibrium ground state is
+described by a $k$RSB solution, the complexity at the ground state is
+$(k-1)$RSB. This can be intuitively understood by considering the difference
+between measuring overlaps between equilibrium \emph{states} and stationary
+\emph{points}. For states, the finest level of the hierarchical description
+gives the typical overlap between two points drawn from the same state, which
+has some distribution about the ground state at nonzero temperature. For
+points, this finest level does not exist.
+
+In general, solving the saddle-point equations for the parameters of the three
+replica matrices is challenging. Unlike the equilibrium case, the solution is
+not extremal, and so minimization methods cannot be used. However, the line of
+simple `supersymmetric' solutions offers a convenient foothold: starting from
+one of these solutions, the parameters $E$ and $\mu$ can be slowly varied to
+find the complexity everywhere. This is how the data in what follows was produced.
+
+\begin{figure}
+  \centering
+  \hspace{-1em}
+  \includegraphics[width=\columnwidth]{figs/316_complexity_contour_1_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter_legend.pdf}
+
+  \caption{
+    Complexity of the $3+16$ model in the energy $E$ and stability $\mu$
+    plane. Solid lines show the prediction of 1RSB complexity, while dashed
+    lines show the prediction of RS complexity. Below the yellow marginal line
+    the complexity counts saddles of increasing index as $\mu$ decreases. Above
+    the yellow marginal line the complexity counts minima of increasing
+    stability as $\mu$ increases.
+  } \label{fig:2rsb.contour}
+\end{figure}
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter_legend.pdf}
+
+  \caption{
+    Detail of the `phases' of the $3+16$ model complexity as a function of
+    energy and stability. Solid lines show the prediction of 1RSB complexity, while dashed
+    lines show the prediction of RS complexity. Above the yellow marginal stability line the
+    complexity counts saddles of fixed index, while below that line it counts
+    minima of fixed stability. The shaded red region shows places where the
+    complexity is described by the 1RSB solution, while the shaded gray region
+    shows places where the complexity is described by the RS solution. In white
+    regions the complexity is zero. Several interesting energies are marked
+    with vertical black lines: the traditional `threshold' $E_\mathrm{th}$
+    where minima become most numerous, the algorithmic threshold
+    $E_\mathrm{alg}$ that bounds the performance of smooth algorithms, and the
+    average energies at the $2$RSB and $1$RSB equilibrium transitions $\langle
+    E\rangle_2$ and $\langle E\rangle_1$, respectively. Though the figure is
+    suggestive, $E_\mathrm{alg}$ lies at slightly lower energy than the termination of the RS
+    -- 1RSB transition line.
+  } \label{fig:2rsb.phases}
+\end{figure}
+
+For the first example, we study a model whose complexity has the simplest
+replica symmetry breaking scheme, 1RSB. By choosing a covariance $f$ as the sum
+of polynomials with well-separated powers, one develops 2RSB in equilibrium.
+This should correspond to 1RSB in the complexity. We take
+\begin{equation}
+  f(q)=\frac12\left(q^3+\frac1{16}q^{16}\right)
+\end{equation}
+established to have a 2RSB ground state \cite{Crisanti_2011_Statistical}.
+With this covariance, the model sees a replica symmetric to 1RSB transition at
+$\beta_1=1.70615\ldots$ and a 1RSB to 2RSB transition at
+$\beta_2=6.02198\ldots$. The typical equilibrium energies at these phase
+transitions are listed in Table~\ref{tab:energies}.
+
+\begin{table}
+  \begin{tabular}{l|cc}
+    & $3+16$ & $2+4$ \\\hline\hline
+    $\langle E\rangle_\infty$ &---& $-0.531\,25\hphantom{1\,111\dots}$ \\
+    $\hphantom{\langle}E_\mathrm{max}$ &     $-0.886\,029\,051\dots$ & $-1.039\,701\,412\dots$\\
+    $\langle E\rangle_1$ & $-0.906\,391\,055\dots$ & ---\\
+    $\langle E\rangle_2$ & $-1.195\,531\,881\dots$ & ---\\
+    $\hphantom{\langle}E_\mathrm{dom}$ & $-1.273\,886\,852\dots$ & $-1.056\,6\hphantom{11\,111\dots}$\\
+    $\hphantom{\langle}E_\mathrm{alg}$ & $-1.275\,140\,128\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{th}$ & $-1.287\,575\,114\dots$ & $-1.059\,384\,319\ldots$\\
+    $\hphantom{\langle}E_\mathrm{m}$ & $-1.287\,605\,527\ldots$ & $-1.059\,384\,319\ldots$ \\
+    $\hphantom{\langle}E_0$ & $-1.287\,605\,530\ldots$ & $-1.059\,384\,319\ldots$\\\hline
+  \end{tabular}
+  \caption{
+    Landmark energies of the equilibrium and complexity problems for the two
+    models studied. $\langle E\rangle_1$, $\langle E\rangle_2$ and $\langle
+    E\rangle_\infty$ are the average energies in equilibrium at the RS--1RSB,
+    1RSB--2RSB, and RS--FRSB transitions, respectively. $E_\mathrm{max}$ is the
+    highest energy at which any stationary points are described by a RSB
+    complexity. $E_\mathrm{dom}$ is the energy at which dominant stationary
+    points have an RSB complexity. $E_\mathrm{alg}$ is the algorithmic
+    threshold below which smooth algorithms cannot go. $E_\mathrm{th}$ is the
+    traditional threshold energy, defined by the energy at which marginal
+    minima become most common. $E_\mathrm m$ is the lowest energy at which
+    saddles or marginal minima are found. $E_0$ is the ground state energy.
+  } \label{tab:energies}
+\end{table}
+
+In this model, the RS complexity gives an inconsistent answer for the
+complexity of the ground state, predicting that the complexity of minima
+vanishes at a higher energy than the complexity of saddles, with both at a
+lower energy than the equilibrium ground state. The 1RSB complexity resolves
+these problems, shown in Fig.~\ref{fig:2rsb.contour}. It predicts the same ground state as equilibrium and with a
+ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. It predicts that
+the complexity of marginal minima (and therefore all saddles) vanishes at
+$E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become
+dominant over minima at a higher energy $E_\mathrm{th}$. The 1RSB complexity
+transitions to a RS description for dominant stationary points at an energy
+$E_\mathrm{dom}$. The highest energy for which the 1RSB description exists is
+$E_\mathrm{max}$. The numeric values for all these energies are listed in
+Table~\ref{tab:energies}.
+
+For minima, the complexity does
+not inherit a 1RSB description until the energy is with in a close vicinity of
+the ground state. On the other hand, for high-index saddles the complexity
+becomes described by 1RSB at quite high energies. This suggests that when
+sampling a landscape at high energies, high index saddles may show a sign of
+replica symmetry breaking when minima or inherent states do not.
+
+Fig.~\ref{fig:2rsb.phases} shows a different detail of the complexity in the
+vicinity of the ground state, now as functions of the energy difference and
+stability difference from the ground state. Several of the landmark energies
+described above are plotted, alongside the boundaries between the `phases.'
+Though $E_\mathrm{alg}$ looks quite close to the energy at which dominant
+saddles transition from 1RSB to RS, they differ by roughly $10^{-3}$, as
+evidenced by the numbers cited above. Likewise, though $\langle E\rangle_1$
+looks very close to $E_\mathrm{max}$, where the 1RSB transition line
+terminates, they too differ. The fact that $E_\mathrm{alg}$ is very slightly
+below the place where most saddle transition to 1RSB is suggestive; we
+speculate that an analysis of the typical minima connected to these saddles by
+downward trajectories will coincide with the algorithmic limit. An analysis of
+the typical nearby minima or the typical downward trajectories from these
+saddles at 1RSB is warranted \cite{Ros_2019_Complex, Ros_2021_Dynamical}. Also
+notable is that $E_\mathrm{alg}$ is at a significantly higher energy than
+$E_\mathrm{th}$; according to the theory, optimal smooth algorithms in this
+model stall in a place where minima are exponentially subdominant.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=\columnwidth]{figs/24_phases_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/24_detail_letter_legend.pdf}
+  \caption{
+    `Phases' of the complexity for the $2+4$ model in the energy $E$ and
+    stability $\mu$ plane. Solid lines show the prediction of 1RSB complexity,
+    while dashed lines show the prediction of RS complexity. The region shaded
+    gray shows where the RS solution is correct, while the region shaded red
+    shows that where the FRSB solution is correct. The white region shows where
+    the complexity is zero.
+  } \label{fig:frsb.phases}
+\end{figure}
+
+If the covariance $f$ is chosen to be concave, then one develops FRSB in equilibrium. To this purpose, we choose
+\begin{equation}
+  f(q)=\frac12\left(q^2+\frac1{16}q^4\right),
+\end{equation}
+also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. Because the ground state is FRSB, for this model $E_0=E_\mathrm{alg}=E_\mathrm{th}=E_\mathrm m$.
+In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty$, also in Table~\ref{tab:energies}.
+
+Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model,
+computed using finite-$k$ RSB approximations. Notably, the phase boundary
+predicted by a perturbative expansion correctly predicts where the finite
+$k$RSB approximations terminate. Like the 1RSB model in the previous
+subsection, this phase boundary is oriented such that very few, low energy,
+minima are described by a FRSB solution, while relatively high energy saddles
+of high index are also. Again, this suggests that studying the mutual
+distribution of high-index saddle points might give insight into lower-energy
+symmetry breaking in more general contexts.
+
+We have used our solution for mean-field complexity to explore how hierarchical
+RSB in equilibrium corresponds to analogous hierarchical structure in the
+energy landscape. In the examples we studied, a relative minority of energy
+minima are distributed in a nontrivial way, corresponding to the lowest energy
+densities. On the other hand, very high-index saddles begin exhibit RSB at much
+higher energy densities, on the order of the energy densities associated with
+RSB transitions in equilibrium. More wore is necessary to explore this
+connection, as well as whether a purely \emph{geometric} explanation can be
+made for the algorithmic threshold. Applying this method to the most realistic
+RSB scenario for structural glasses, the so-called 1FRSB which has features of
+both 1RSB and FRSB, might yield insights about signatures that should be
+present in the landscape.
+
+\paragraph{Acknowledgements}
+The authors would like to thank Valentina Ros for helpful discussions.
+
+\paragraph{Funding information}
+JK-D and JK are supported by the Simons Foundation Grant No.~454943.
+
+\bibliography{frsb_kac-rice}
+
+\end{document}