From 8a0c4b50f2ac99b3c1ab12871fa1811c4d01d569 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Mar 2022 15:43:13 +0200
Subject: Added paragraph to the numerics section.

---
 stokes.tex | 11 +++++++++++
 1 file changed, 11 insertions(+)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 7564b6b..c6f603f 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1845,6 +1845,17 @@ gapped minima are unlikely to see Stokes lines.
   } \label{fig:numeric.angle.gap}
 \end{figure}
 
+We can also see that as the empirical gap is increased, Stokes points tend to
+occur at very large phases. This can be seen for $N=32$ in
+Fig.~\ref{fig:numeric.angle.gap}, which shows the probability distribution of
+Stokes lines discovered as a function of phase $|\theta|$ necessary to reach
+them. The curves are broken into sets representing different bins of the
+empirical gap $|\lambda_\textrm{min}|$. As the empirical gap grows, Stokes
+points become depleted around small phases and concentrate on very large ones.
+This supports the idea that around the gapped minima, Stokes points will be
+concentrated at phases that are nearly $180^\circ$, where the two-replica
+calculation shows that almost all of their nearest neighbors will lie.
+
 \subsection{Pure {\it p}-spin: is analytic continuation possible?}
 
 After this work, one is motivated to ask: can analytic continuation be done in
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