From 82b077333072bafbfa9272fd1d075edf790aa658 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 28 Mar 2022 15:50:10 +0200
Subject: Added discussion to the p-spin complexity subsection.

---
 stokes.tex | 45 ++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 38 insertions(+), 7 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 4a5f3d5..f01a4b1 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1349,9 +1349,9 @@ $\gamma^\dagger\gamma$, $\eta^\dagger\gamma$, and $\eta^T\gamma$. The saddle
 point equations can be used to eliminate all but one of these, the `radius'
 like term $z^\dagger z$. When combined with the constraint, this term can be
 related directly to the magnitude of the imaginary part of $z$, since
-$z^\dagger z=x^Tx+y^Ty=N+2y^Ty$. We therefore define $Y=\frac1N(z^\dagger
+$z^\dagger z=x^Tx+y^Ty=N+2y^Ty$. We therefore define $(\operatorname{Im}s)^2=\frac1N(z^\dagger
 z-N)$, the specific measure of the distance into the complex plane from the
-real sphere. The complexity can then be written
+real sphere. The complexity can then be written in terms of $r=z^\dagger z/N$ as
 \begin{equation}
   \Sigma
   =
@@ -1361,6 +1361,7 @@ real sphere. The complexity can then be written
   -\frac{(\operatorname{Re}\epsilon)^2}{R_+^2}-\frac{(\operatorname{Im}\epsilon)^2}{R_-^2}
   +I_p(\epsilon/\epsilon_\mathrm{th})
 \end{equation}
+where
 \begin{equation}
   R_\pm^2=\frac{p-1}2\frac{(r^{p-2}\pm1)\left[
       r^{2(p-1)}\pm(p-1)r^{p-2}(r^2-1)-1
@@ -1368,7 +1369,7 @@ real sphere. The complexity can then be written
     1+r^{2(p-2)}\left[p(p-2)(r^2-1)-1\right]
   }
 \end{equation}
-$I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$.
+and the function $I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$ and
 \begin{equation}
   \eqalign{
     I_p(u)
@@ -1384,14 +1385,12 @@ $I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$.
     \right)
   }
 \end{equation}
-where the branch of the square roots are chosen such that the real part of the
-root has the opposite sign as the real part of $u$, e.g., if
+otherwise. The branch of the square roots are chosen such that the real part of
+the root has the opposite sign as the real part of $u$, e.g., if
 $\operatorname{Re}u<0$ then $\operatorname{Re}\sqrt{u^2-1}>0$. If the real part
 is zero, then the sign is taken so that the imaginary part of the root has the
 opposite sign of the imaginary part of $u$.
 
-\cite{Auffinger_2012_Random}
-
 \begin{figure}
   \hspace{4pc}
   \includegraphics{figs/re_complexity.pdf}
@@ -1408,6 +1407,14 @@ opposite sign of the imaginary part of $u$.
   } \label{fig:p-spin.complexity}
 \end{figure}
 
+Contours of this complexity for the pure 3-spin are plotted in
+Fig.~\ref{fig:p-spin.complexity} for pure and imaginary energy. The thick black
+line shows the contour of zero complexity, where stationary points are no
+longer found at large $N$. As the magnitude of the imaginary part of the spin
+taken greater, more stationary points are found, and at a wider array of
+energies. This is also true in other directions into the complex energy plane,
+where the story is qualitatively the same.
+
 \begin{figure}
   \hspace{2pc}
   \includegraphics{figs/ground_complexity.pdf}
@@ -1425,6 +1432,30 @@ opposite sign of the imaginary part of $u$.
   } \label{fig:ground.complexity}
 \end{figure}
 
+Something more interesting is revealed if we zoom in on the complexity around
+the ground state, shown in Fig.~\ref{fig:ground.complexity}. Here, the region
+where most stationary points have a gapped hessian is shaded. The line
+separating gapped from ungapped distribution corresponds to the threshold
+energy $\epsilon_\mathrm{th}$ in the limit of $(\operatorname{Im}s)^2\to0$.
+Above the threshold, the limit of the complexity to zero imaginary component
+(or equivalently $r\to1$) also approaches the real complexity, plotted under
+the horizontal axis. However, below the threshold this is no longer the case:
+here the limit of $(\operatorname{Im}s)^2\to0$ of the complexity of complex
+stationary points corresponds to the complexity of \emph{rank one saddles} in
+the real problem, and their complexity becomes zero at $\epsilon_1$, where the
+complexity of rank one saddles becomes zero \cite{Auffinger_2012_Random}.
+
+There are several interesting features of the complexity. First is this
+inequivalence between the real complexity and the limit of the complex
+complexity to zero complex part. It implies, among other things, a desert of
+stationary points in the complex plan surrounding the lowest minima, something
+we shall see more explicitly in the next section. Second, there is only a small
+collection of stationary points that appear with positive complexity and a
+gapped spectrum: the small region in Fig.~\ref{fig:ground.complexity} that is
+both to the right of the thick line and brightly shaded. We suspect that these
+are the only stationary points that are somewhat protected from participation
+in Stokes points.
+
 \subsection{Pure \textit{p}-spin: where are my neighbors?}
 
 The problem of counting the density of Stokes points in an analytic
-- 
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