From 84dff2e9f7ed4da29f02ab90105bb85f9a9f6fce Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Sun, 22 Jan 2023 15:21:21 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice_letter.tex | 124 +++++++++++++++++++++++++++++------------------
 1 file changed, 77 insertions(+), 47 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

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
-- 
cgit v1.2.3-70-g09d2


From a8850fdb22cec36eb0f4981c4ff66163b34e990d Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Mon, 23 Jan 2023 10:36:35 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice_letter.tex | 14 ++++++++------
 response.txt             |  9 +++++----
 2 files changed, 13 insertions(+), 10 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 27025e7..f41e8ca 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -26,12 +26,14 @@
 \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 representative
-  models. Including
-  replica symmetry breaking uncovers s the full  organization of saddles in terms of their energies and stabilities encountered in generic models.
+ {\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  organization of saddles in terms of their energies and 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.
diff --git a/response.txt b/response.txt
index 058a6f8..424aec8 100644
--- a/response.txt
+++ b/response.txt
@@ -40,10 +40,11 @@
 
 \opening{Dear Dr.~Curatolo,}
 
-
-Because neither referee addressed (or criticized)  the scientific content of our paper,
-nor substantively addressed its presentation, we have only modified
-the manuscripts in order to highlight the importance of having a full solution. 
+Enclosed please find a revised manuscript.
+Neither referee addressed (or criticized)  the scientific content of our paper,
+nor substantively addressed its presentation. We have followed their comments
+in the direction of  highlighting the importance of having a full solution. In particular
+we have emphasized that going to the full replica treatment 
 
 
 We have thus added the paragraph:
-- 
cgit v1.2.3-70-g09d2


From 148b89a29e563e104f6681716d23a2111bebf4a5 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Mon, 23 Jan 2023 10:40:15 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice_letter.tex | 2 +-
 response.txt             | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index f41e8ca..b6e7b8b 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -32,7 +32,7 @@
 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  organization of saddles in terms of their energies and stabilities, a structure that  is lost in the annealed approximation. This structure should be a guide for the identification
+  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
diff --git a/response.txt b/response.txt
index 1a830e8..1709e15 100644
--- a/response.txt
+++ b/response.txt
@@ -41,7 +41,7 @@
 \opening{Dear Dr.~Curatolo,}
 
 Enclosed please find a revised manuscript.
-Neither referee addressed (or criticized)  the scientific content of our paper,
+Neither referee  criticized  the scientific content of our paper,
 nor substantively addressed its presentation. We have followed their comments
 in the direction of  highlighting the importance of having a full solution. In particular
 we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment.
-- 
cgit v1.2.3-70-g09d2


From 87cc37c5c6cd8d1d3a29d1cda94ad9e1f9ec790f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 23 Jan 2023 16:38:05 +0100
Subject: Some rewriting.

---
 frsb_kac-rice_letter.tex | 141 ++++++++++++++++++++---------------------------
 1 file changed, 59 insertions(+), 82 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

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
-- 
cgit v1.2.3-70-g09d2


From 5c203019791f877f16c7f4819693c8f28b472310 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 26 Jan 2023 15:58:54 +0100
Subject: Some writing edits.

---
 frsb_kac-rice_letter.tex | 89 ++++++++++++++++++++++--------------------------
 1 file changed, 40 insertions(+), 49 deletions(-)

(limited to 'frsb_kac-rice_letter.tex')

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 5c29523..c7ef913 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -29,13 +29,14 @@
   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.
+  models. The annealed approximation is well understood, but is generally not
+  exact. Here we describe 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
@@ -67,58 +68,48 @@ relatively small subset of models, including most notably the (pure) $p$-spin sp
 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 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}.
+primarily a matter of {\em accuracy}. Basic structural questions are
+omitted in the approximate `annealed' solution. What is lost is the nature of
+the stationary points at a given energy level: 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 if one hopes to correctly describe more complex
+objects such as barrier crossing (which barriers?) \cite{Ros_2019_Complexity,
+Ros_2021_Dynamical} or the fate of long-time dynamics (that end in which kind
+of 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 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.
-
-}
-
-{\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.
+saddles, and in particular the lowest ones. Using a
+general form for the solution detailed in a companion article \cite{Kent-Dobias_2022_How}, we describe the
+structure of landscapes with a 1RSB complexity and a full RSB complexity. The interpretation of a Parisi
+ansatz itself must be recast to make sense of the new order parameters
+involved.
+
+For simplicity we concentrate 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 and it suffices
+to apply the same computation to the Thouless--Anderson--Palmer
+\cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the
+energy. We do not expect new features or technical complications to arise.
 
 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 $s_1,s_2\in S^{N-1}$,
+spherical models, which exhibit a zoo of disordered phases. These models can be
+defined by drawing a random Hamiltonian $H$ from a distribution of isotropic
+Gaussian fields defined on the $N-1$ sphere. Isotropy implies that the
+covariance in energies between two configurations depends on only their dot
+product (or overlap), so for $\mathbf s_1,\mathbf s_2\in
+S^{N-1}$,
 \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}
-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 $f(q)=\frac12q^p$.
+where $f$ is a function with positive coefficients. The overbar will always
+denote an average over the functions $H$. The choice of function $f$ uniquely
+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
-- 
cgit v1.2.3-70-g09d2

