From a58f53debfe39ac7a098defef95e170cbcd360c0 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 Dec 2020 19:26:02 +0100
Subject: Added paragraph discussing the threshold.

---
 bezout.tex | 48 ++++++++++++++++++++++++++++++++++--------------
 1 file changed, 34 insertions(+), 14 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 7fbe3cc..c586908 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -334,11 +334,13 @@ Notice that the limit of this expression as $a\to\infty$ corresponds with
 plotted as a function of $a$ for several values of $\kappa$ in
 Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to
 negative infinity as $a\to1$, i.e., as the spins are restricted to the reals.
-However, when the result is analytically continued to $\kappa=1$ (which
-corresponds to real $J$) something novel occurs: the complexity has a finite
-value at $a=1$.  Since the $a$-dependence gives a cumulative count, this
-implies a $\delta$-function density of critical points along the line $y=0$.
-The number of critical points contained within is
+This is natural, given that the $y$ contribution to the volume shrinks to zero
+as that of an $N$-dimensional sphere $\sim(a-1)^N$.  However, when the result
+is analytically continued to $\kappa=1$ (which corresponds to real $J$)
+something novel occurs: the complexity has a finite value at $a=1$.  Since the
+$a$-dependence gives a cumulative count, this implies a $\delta$-function
+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},
@@ -358,15 +360,17 @@ limit as $\epsilon$ is varied.
   } \label{fig:complexity}
 \end{figure}
 
-{\color{teal} {\bf somewhere else}
-
-
-{\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a gap of  $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
-For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation
-is more interesting. In the range of  values of $\Re H_0$ where there are exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. 
-Another remarkable feature of the limit $\kappa=1$ is that there is still a gap without solutions around
-`deep' real energies where there is no real solution. A moment's thought tells us that this is a necessity: otherwise a small perturbation of the $J$'s could produce a real, unusually deep solution for the real problem, in a region where we expect this not to happen.
-}
+These qualitative features carry over to nonzero $\epsilon$. In
+Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$
+close to one for which there are no solutions. For the case $\kappa=1$ -- the
+analytic continuation to the usual real computation -- the situation is more
+interesting. In the range of energies where there are real solutions this gap
+closes, and this is only possible if the density of solutions diverges at
+$a=1$.  Another remarkable feature of the limit $\kappa=1$ is that there is
+still a gap without solutions around `deep' real energies where there is no
+real solution. A moment's thought tells us that this is a necessity: otherwise
+a small perturbation of the $J$'s could produce a real, unusually deep solution
+for the real problem, in a region where we expect this not to happen.
 
 \begin{figure}[htpb]
   \centering
@@ -378,6 +382,22 @@ Another remarkable feature of the limit $\kappa=1$ is that there is still a gap
   } \label{fig:desert}
 \end{figure}
 
+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
+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.,
+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.
+
 \begin{figure}[htpb]
   \centering
 
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