Update on Overleaf.

1 files changed, 77 insertions, 47 deletions

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 4fb22e7..27025e7 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -29,11 +29,12 @@
   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.
+  as it should. Using this solution, we count the stationary points of two representative
+  models. Including
+  replica symmetry breaking uncovers s the full  organization of saddles in terms of their energies and stabilities encountered in generic models.
+  %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
@@ -45,48 +46,80 @@ 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.
+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}.
+
+{\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?). 
+
+
+
+
+
+
+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.
+
+}
+
+{\color{blue}
+  For simplicity we have concentrated here on the energy, rather
+than  {\em free-energy} landscape, although the latter is sometimes
+more appropriate. From the technical point of view, this makes no fundamental difference, it suffices 
+to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical 
+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.}.
-
-We study the mixed $p$-spin spherical models, with Hamiltonian
+%\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}
 \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
+ $\mathbf s\in\mathbb R^N$ 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
@@ -95,19 +128,16 @@ 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,
+ the so-called  `TAP' complexity of pure models \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
+Giardina_2005_Supersymmetry}, and more recently mixed models \cite{Folena_2020_Rethinking} without RSB \cite{Auffinger_2012_Random,
+Auffinger_2013_Complexity, BenArous_2019_Geometry}. 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
+{\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}
@@ -120,7 +150,7 @@ where $f$ is defined by the series
 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 `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