Some rewriting.

1 files changed, 59 insertions, 82 deletions

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index b6e7b8b..5c29523 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -26,17 +26,16 @@
 \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
 
 \begin{abstract}
- {\color{red}  Complex landscapes are defined as those having a proliferation of saddle points.
-  The question of their  number and organization has been the object of long-standing attention, in particular centered around Gaussian mean-field potentials,
-  which includes glass and spin glass models.
-The annealed approximation is by now well understood, but is exact for  a restricted subset of these problems. Here we derive the  exact quenched 
-solution for the general case, which incorporates Parisi's solution for the ground state,
-  as it should. More importantly,  including
-  replica symmetry breaking uncovers  the full  distribution of saddles at given  energy in terms of  their stabilities, a structure that  is lost in the annealed approximation. This structure should be a guide for the identification
-  of relevant activated processes in relaxational or driven dynamics.}
-  %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.
+  Complex landscapes are defined by their many saddle points. Determining their
+  number and organization is a long-standing problem, in particular for
+  tractable Gaussian mean-field potentials, which include glass and spin glass
+  models. The annealed approximation is well understood, but is generally not exact. Here we derive the exact
+  quenched solution for the general case, which incorporates Parisi's solution
+  for the ground state, as it should. More importantly, the quenched solution
+  correctly uncovers the full distribution of saddles at a given energy, a
+  structure that is lost in the annealed approximation. This structure should
+  be a guide for the accurate identification of the relevant activated
+  processes in relaxational or driven dynamics.
 \end{abstract}
 
 \maketitle
@@ -48,49 +47,47 @@ 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  exist in very high dimensions.
-Many  interesting versions of the problem have been treated, and the subject has
-evolved into an active field of probability
-theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity,
-BenArous_2019_Geometry} and  has been applied to
-energy functions inspired by molecular biology, evolution, and machine learning
-\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}.
-
-
-
-
-
-The computation of the number of metastable states in this setting
-was pioneered  forty years ago by Bray and Moore
-\cite{Bray_1980_Metastable}, who proposed the first calculation for the
-Sherrington--Kirkpatrick model, in one of the
-first applications of any replica symmetry breaking (RSB) scheme. As was clear
-from the later results by Parisi \cite{Parisi_1979_Infinite},  their  result was not exact, and the problem has
-been open ever since. To date the program of
-computing the statistics of stationary points---minima, saddle points, and
-maxima---of  mean-field complex landscapes has been only carried out in an exact form for a relatively small subset of
-models, including most notably the (pure) $p$-spin model ($p>2$)
-\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}.
+complex landscapes  exist in very high dimensions. Many  interesting versions
+of the problem have been treated, and the subject has evolved into an active
+field of probability theory \cite{Auffinger_2012_Random,
+  Auffinger_2013_Complexity, BenArous_2019_Geometry} and  has been applied to
+  energy functions inspired by molecular biology, evolution, and machine
+  learning \cite{Maillard_2020_Landscape, Ros_2019_Complex,
+  Altieri_2021_Properties}.
+
+The computation of the number of metastable states in such a landscape was
+pioneered forty years ago by Bray and Moore \cite{Bray_1980_Metastable} on the
+Sherrington--Kirkpatrick (SK) model in one of the first applications of any
+replica symmetry breaking (RSB) scheme. As was clear from the later results by
+Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the
+problem has been open ever since. To date the program of computing the
+statistics of stationary points---minima, saddle points, and maxima---of
+mean-field complex landscapes has been only carried out in an exact form for a
+relatively small subset of models, including most notably the (pure) $p$-spin spherical
+model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer,
+Cavagna_1997_An, Cavagna_1998_Stationary}.
 
 {\color{red} 
-Having  a full, exact  (`quenched') solution of the generic problem  is not 
-primarily a matter of {\em accuracy}.
-Very basic structural questions are omitted in the approximate `annealed' solution. What is lost  is  the nature, at any given
-energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with  an exponentially small number of saddles, or
--- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? 
-These questions need to be answered for the understanding of the relevance of more complex objects such as
-barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics
-(which end in what kind of  target states?). 
-
-
-
-
-
+Having  a full, exact  (`quenched') solution of the generic problem  is not
+primarily a matter of {\em accuracy}. Very basic structural questions are
+omitted in the approximate `annealed' solution. What is lost  is  the nature,
+at any given energy (or free energy) level, of the stationary points in a
+generic energy function: at low energies are they basically all minima, with an
+exponentially small number of saddles, or -- as we show here -- do they consist
+of a mixture of saddles whose index -- the number of unstable directions -- is
+a smoothly distributed number? These questions need to be answered for the
+understanding of the relevance of more complex objects such as barrier crossing
+(which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate
+of long-time dynamics (which end in what kind of  target states?). In fact, we show that the
+state of dynamics in generic cases is limited to energies \emph{at which saddles
+are exponentially more numerous than minima}.
 
 In this paper we present what we argue is the general replica ansatz for the
 number of stationary points of generic mean-field models, which we expect to
-include the Sherrington--Kirkpatrick model. This allows us
-to clarify the rich structure of all the saddles, and in particular the lowest ones. The interpretation of a Parisi ansatz itself, in this context must be recast in a way that makes sense for the order parameters involved.
+include the SK model. This allows us to clarify the rich structure of all the
+saddles, and in particular the lowest ones. The interpretation of a Parisi
+ansatz itself, in this context must be recast in a way that makes sense for the
+order parameters involved.
 
 }
 
@@ -108,25 +105,20 @@ complications arise.
 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.}.
-
-For definiteness, we consider the standard example of 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}
+structure of landscapes with a 1RSB complexity and a full RSB complexity.
+
+For definiteness, we consider the standard example of the mixed $p$-spin
+spherical models, which exhibit a zoo of orders and phases. These models can be
+defined by taking a random Gaussian Hamiltonian $H$ defined on the $N-1$ sphere
+and with a covariance that depends on only the dot product (or overlap) between
+two configurations. For $s_1,s_2\in S^{N-1}$,
+\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}
- $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere
-$\{|\mathbf s\|^2=N\}$. 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.
+where $f$ is a function with positive coefficients. This uniquely defines the
+distribution over Hamiltonians $H$. The overbar will always denote an average
+over the functions $H$. The choice of function $f$ fixes the model. For
+instance, the `pure' $p$-spin models have $f(q)=\frac12q^p$.
 
 The complexity of the $p$-spin models has been extensively studied by
 physicists and mathematicians. Among physicists, the bulk of work has been on
@@ -139,21 +131,6 @@ 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}.
 
-{\color{green} {\bf eliminate?} 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