Update on Overleaf.

1 files changed, 18 insertions, 18 deletions

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index c7ef913..84da6cd 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -26,10 +26,10 @@
 \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}
 
 \begin{abstract}
-  Complex landscapes are defined by their many saddle points. Determining their
+  Complex landscapes are characterized 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
+  models. The annealed approximation is well understood, but is generically 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
@@ -51,16 +51,16 @@ 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
+  Auffinger_2013_Complexity, BenArous_2019_Geometry} that 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
+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
+Parisi \cite{Parisi_1979_Infinite}, their result was only approximate, 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
@@ -68,7 +68,7 @@ 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}.
 
-Having  a full, exact  (`quenched') solution of the generic problem  is not
+Having  a full `quenched' solution of the generic problem  is not
 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
@@ -77,7 +77,7 @@ 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
+Ros_2021_Dynamical} or the fate of long-time dynamics (that targets which kind
 of states?).
 
 In this paper we present what we argue is the general replica ansatz for the
@@ -117,20 +117,20 @@ physicists and mathematicians. Among physicists, the bulk of work has been on
 Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An,
 Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
 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
+Auffinger_2013_Complexity, BenArous_2019_Geometry}. 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 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 family of spherical models thus defined is  rich,  by varying the
+covariance $f$  any hierarchical structure can be found in
+equilibrium. Because of the correspondence between the ground state complexity
+and the equilibrium entropy, any hierarchical structure in 
+equilibrium should be reflected in the computation.
 
 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
+\cite{Kac_1943_On, Rice_1939_The}. It is given by
 \begin{equation}
   \begin{aligned}
     \mathcal N(E, \mu)
@@ -156,9 +156,9 @@ is a Wigner semicircle of radius $\mu_\mathrm m=\sqrt{4f''(1)}$ centered at $\mu
 $\mu>\mu_\mathrm m$, 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).
+index fixed to within order one (fixed macroscopic index).
 
-It's worth reviewing the complexity for the best-studied case of the pure model
+It is 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
@@ -346,7 +346,7 @@ 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
+ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. Also, 
 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
@@ -356,7 +356,7 @@ $E_\mathrm{max}$. 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
+not inherit a 1RSB description until the energy is within 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