Some writing edits.

1 files changed, 40 insertions, 49 deletions

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