Update on Overleaf.

1 files changed, 16 insertions, 7 deletions

diff --git a/bezout.tex b/bezout.tex
index b4b52e0..06ccac7 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -118,7 +118,9 @@ of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ an
 
 Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical  point of $\Im H_0$. The number of
 critical points of $H$ is therefore the number of critical points of
-$\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
+$\mathop{\mathrm{Re}}H$. From each critical point emerges a gradient line of $\Re H_0$, which is also one of constant phase $\Im H_0=const$.
+
+Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
 interpreted as a real function of $2N$ real variables. The number of critical
 points it has is given by the usual Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
@@ -381,7 +383,7 @@ density of critical points along the line $y=0$.  The number of critical points
 contained within is
 \begin{equation}
   \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a)
-  =(p-1)^{N/2},
+  \sim (p-1)^{N/2},
 \end{equation}
 the square root of \eqref{eq:bezout} and precisely the number of critical
 points of the real pure spherical $p$-spin model. (note the role of conjugation
@@ -424,16 +426,15 @@ for the real problem, in a region where we expect this not to happen.
 The relationship between the threshold and ground state energies is richer than
 in the real case. In Fig.~\ref{fig:eggs} these are shown in the
 complex-$\epsilon$ plane for several examples. Depending on the parameters, the
-threshold always come at smaller magnitude than the ground state, or always
-come at larger magnitude than the ground state, or change relative size as a
+threshold line always come at smaller magnitude than the ground state, or always
+come at larger magnitude than the ground state, or cross as a
 function of complex argument. For sufficiently large $a$ the threshold always
 comes at larger magnitude than the ground state. If this were to happen in the
 real case, it would likely imply our replica symmetric computation is unstable,
 as having the ground state above the threshold would imply a ground state
 Hessian with many negative eigenvalues, a contradiction with the notion of a
-ground state. However, this is not a contradiction in the complex case, since
-there cannot be minima of a complex function and the ground state therefore
-takes on a different meaning. The relationship between the threshold, i.e.,
+ground state. However, this is not an obvious  contradiction in the complex case. 
+The relationship between the threshold, i.e.,
 where the gap appears, and the dynamics of, e.g., a minimization algorithm or
 physical dynamics, are a problem we hope to address in future work.
 
@@ -453,6 +454,14 @@ physical dynamics, are a problem we hope to address in future work.
     any energy.
   } \label{fig:eggs}
 \end{figure}
+\section{Perspectives}
+
+This paper provides a first step for the study of a complex landscape with complex variables. The next obvious one
+is to study the topology of the critical points and the lines of constant phase.
+We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it 
+does in the real case.
+
+
 
 \begin{acknowledgments}
   JK-D and JK are supported by the Simons Foundation Grant No.~454943.