Small changes of Jorge's edits.

1 files changed, 9 insertions, 9 deletions

diff --git a/bezout.tex b/bezout.tex
index f596a89..4b3bd19 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -86,8 +86,8 @@ defining and analyzing path integrals with complex action (see
 \cite{Witten_2010_A, Witten_2011_Analytic}), and as a useful palliative for the
 sign problem \cite{Cristoforetti_2012_New, Tanizaki_2017_Gradient,
 Scorzato_2016_The}.  In order to do this correctly, features of landscape
-of the action in complex space--- such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here
-we propose to follow standard the strategy of computer science of understanding the
+of the action in complex space---such as the relative position of saddles and the existence of Stokes lines joining them---must be understood. Such landscapes are in general not random: here
+we follow the standard strategy of computer science by understanding the
 generic features of random instances, expecting that this sheds light on
 practical, nonrandom problems.
 
@@ -106,7 +106,7 @@ holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
 this, save for a single anomaly: the value of the energy and its gradient at
 any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
 H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous direction
-thus best be forbidden, and the constraint surface $z^Tz=N$ does precisely this.
+should thus be forbidden, and the constraint surface $z^Tz=N$ accomplishes this.
 
 Second, taking the constraint to be the level set of a holomorphic function
 means the resulting configuration space is a \emph{bone fide} complex manifold,
@@ -116,13 +116,13 @@ z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
 Imposing the constraint with  a holomorphic function
-makes the resulting configuration space is a \emph{bone fide} complex manifold, which is, as we mentioned, the
+makes the resulting configuration space a \emph{bone fide} complex manifold, which is, as we mentioned, the
 situation we wish to model. The same cannot be said for the space defined by $z^\dagger
-z=N$, which is topologically the $(2N-1)$-sphere,  does not admit a complex
+z=N$, which is topologically the $(2N-1)$-sphere, does not admit a complex
 structure, and thus yields a trivial structure of saddles.
-However, we will introduce the domains of
-`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$, as a device to classify saddles.   We shall see that the
-`radius' $r$ and its upper bound $R$ prove to be insightful knobs in our present
+However, we will introduce the bound $r^2\equiv z^\dagger z/N\leq R^2$
+on the `radius' per spin as a device to classify saddles.   We shall see that this
+`radius' $r$ and its upper bound $R$ are insightful knobs in our present
 problem, revealing structure as they are varied. Note that taking $R=1$ reduces
 the problem to that of the ordinary $p$-spin.
 
@@ -155,7 +155,7 @@ usual Kac--Rice formula applied to $\operatorname{Re}H$:
 \end{equation}
 This expression is to be averaged over $J$ to give the complexity $\Sigma$ as
 $N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica
-trick. Based on the experience from these problems \cite{Castellani_2005_Spin-glass},  the
+trick. Based on the experience from similar problems \cite{Castellani_2005_Spin-glass},  the
 \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is
 expected to be exact wherever the complexity is positive.