Update on Overleaf.

1 files changed, 14 insertions, 18 deletions

diff --git a/bezout.tex b/bezout.tex
index f050707..f596a89 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -69,7 +69,7 @@ $z\in\mathbb C^N$ and $J$ to be a symmetric tensor whose elements are
 $\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The
 constraint remains $z^Tz=N$.
 
-The motivations for this paper are of two types. On the practical side, there
+The motivations for this paper are of three types. On the practical side, there
 are indeed situations in which complex variables appear naturally in disordered
 problems: such is the case in which the variables are \emph{phases}, as in
 random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used
@@ -80,15 +80,14 @@ plane often uncovers underlying simplicity that is otherwise hidden, shedding
 light on the original real problem, e.g., as in the radius of convergence of a
 series.
 
-Deforming an integral in $N$ real variables to a surface of dimension $N$ in
+Finally, deforming an integral in $N$ real variables to a surface of dimension $N$ in
 $2N$-dimensional complex space has turned out to be necessary for correctly
 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---like the relative position of its
-saddles---must be understood. Such landscapes are in general not random: here
-we propose to follow 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 propose to follow standard the strategy of computer science of understanding the
 generic features of random instances, expecting that this sheds light on
 practical, nonrandom problems.
 
@@ -107,9 +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
-must be neglected, and the constraint surface $z^Tz=N$ is the unique surface
-whose normal is parallel to $z$ and which contains the configuration space of
-the real $p$-spin model as a subspace.
+thus best be forbidden, and the constraint surface $z^Tz=N$ does precisely 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,
@@ -118,14 +115,13 @@ referenced above. The same cannot be said for the space defined by $z^\dagger
 z=N$, which is topologically the $(2N-1)$-sphere and cannot admit a complex
 structure.
 
-A consequence of the constraint is that the model's configuration space is not
-compact, nor is its energy bounded. This is not necessarily problematic, as many
-related problems have similar properties but are concerned with subspaces on
-which the energy is bounded. (In fact, identifying the appropriate subspace
-often requires the study of critical points in the whole space.) Where it might
-become problematic, we introduce an additional constraint that bounds the
-`radius' per spin $r^2\equiv z^\dagger z/N\leq R^2$.  The resulting
-configuration space is a complex manifold with boundary. We shall see that the
+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
+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
+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
 problem, revealing structure as they are varied. Note that taking $R=1$ reduces
 the problem to that of the ordinary $p$-spin.
@@ -159,9 +155,9 @@ 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. In most of the parameter space that we shall study here, the
+trick. Based on the experience from these problems \cite{Castellani_2005_Spin-glass},  the
 \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is
-exact. 
+expected to be exact wherever the complexity is positive.
 
 The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
 a manifestly complex way.  With the Wirtinger derivative