Spelling and abstract.

1 files changed, 11 insertions, 8 deletions

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index ec71085..680cc89 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -25,9 +25,11 @@
 \author{Jaron Kent-Dobias \& Jorge Kurchan}
 \maketitle
 \begin{abstract}
-    We derive the general solution for the computation of stationary points of
+    We derive the general solution for counting the stationary points of
     mean-field complex landscapes. It incorporates Parisi's solution
-    for the ground state, as it should.
+    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.
 \end{abstract}
 
 \section{Introduction}
@@ -410,7 +412,7 @@ the value of the stability $\mu$ in all replicas to the value $\mu^*$.
   \item For $\mu^*<\mu_m$, this amounts to fixing the index density. Since the
     overwhelming majority of saddles have a semicircle distribution, the
     fluctuations are rarer than exponential.
-  \item For the gapped case $\mu^*>\mu_m$, there is a an exponentially small
+  \item For the gapped case $\mu^*>\mu_m$, there is an exponentially small
     probability that $r=1,2,...$ eigenvalues detach from the semicircle in such
     a way that the index is in fact $N {\cal{I}}=r$.  We shall not discuss
     these subextensive index fluctuations in this paper, the interested reader
@@ -697,7 +699,7 @@ $\hat\beta=\tilde\beta$, and $C=\tilde Q$.
 {\em Therefore a $(k-1)$RSB ansatz in
 Kac--Rice will predict the correct ground state energy for a model whose
 equilibrium state at small temperatures is $k$RSB } Moreover, there is an
-exact correspondance between the saddle parameters of each.  If the equilibrium
+exact correspondence between the saddle parameters of each.  If the equilibrium
 is given by a Parisi matrix with parameters $x_1,\ldots,x_k$ and
 $q_1,\ldots,q_k$, then the parameters $\hat\beta$, $r_d$, $d_d$, $\tilde
 x_1,\ldots,\tilde x_{k-1}$, and $c_1,\ldots,c_{k-1}$ for the
@@ -1346,9 +1348,10 @@ global overlap between different states is at most $1/2$.  At $\hat \beta>\hat
 \subsection{\textit{R} and \textit{D}: response functions}
 
 The matrix field $R$ is related to responses of the stationary points to
-perturbations of the tensors $J$. One adds to the Hamiltonian a random term $\varepsilon \tilde H_p = \varepsilon \sum_{i_1,...,i_p} \tilde J_{i_1,...,i_p} s_{i_1}...s_{i_p}$, where the $\tilde J$ are 
-random Gaussian uncorrelated with the $J$'s. 
-The response to these is:
+perturbations of the tensors $J$. One adds to the Hamiltonian a random term
+$\varepsilon \tilde H_p = \varepsilon \sum_{i_1,...,i_p} \tilde J_{i_1,...,i_p}
+s_{i_1}...s_{i_p}$, where the $\tilde J$ are random Gaussian uncorrelated with
+the $J$'s. ] The response to these is:
 \begin{equation}
   \begin{aligned}
   & \overline{  \frac{\partial \langle \tilde H_p \rangle_{\tilde J}  }  {\partial \varepsilon} } 
@@ -1454,7 +1457,7 @@ saddles at this transition point.
 The authors would like to thank Valentina Ros for helpful discussions.
 
 \paragraph{Funding information}
-J K-D and J K are supported by the Simons Foundation Grant No. 454943.
+JK-D and JK are supported by the Simons Foundation Grant No.~454943.
 
 \printbibliography