From ed7025ba62d7e80e859434768b8a256f2320bbeb Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 11 Feb 2022 18:19:54 +0100
Subject: Added discussion of bound on overlap, and a figure illustrating the
 bound.

---
 figs/bound.pdf | Bin 0 -> 30251 bytes
 stokes.tex     |  28 +++++++++++++++++++++++++++-
 2 files changed, 27 insertions(+), 1 deletion(-)
 create mode 100644 figs/bound.pdf

diff --git a/figs/bound.pdf b/figs/bound.pdf
new file mode 100644
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diff --git a/stokes.tex b/stokes.tex
index f2ce388..fa65c10 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1455,7 +1455,33 @@ variable $\delta$.
 
 Though the value of $\delta$ is bounded by $\Delta$ by $|\delta|\leq\Delta$, in
 reality this bound is not the relevant one, because we are confined on the
-manifold $N=z^2$.
+manifold $N=z^2$. This bound is most easily established by returning to a
+$2N$-dimensional real problem, with $x=x_1$ and $z=x_2+iy_2$. The constraint gives $x_2^Ty_2=0$, $x_1^Tx_1=1$, and $x_2^Tx_2=1+y_2^Ty_2$. Then
+\begin{equation}
+  \Delta=1+x_2^Tx_2+y_2^Ty_2-2x_1^Tx_2=2(1+y_2^Ty_2-x_1^Tx_2)
+\end{equation}
+\begin{equation}
+  \Delta=2(1+|y_2|^2-\sqrt{1-|y_2|^2}\cos\theta_{xx})
+\end{equation}
+\begin{eqnarray}
+  \delta&=2-2x_1^Tx_2-2ix_1^Ty_2=2(1-|x_2|\cos\theta_{xx}-i|y_2|\cos\theta_{xy}) \\
+        &=2(1-\sqrt{1-|y_2|^2}\cos\theta_{xx}-i|y_2|\cos\theta_{xy})
+\end{eqnarray}
+$\cos^2\theta_{xy}\leq1-\cos^2\theta_{xx}$
+These equations along with the inequality produce the required bound on $|\delta|$ as a function of $\Delta$ and $\arg\delta$.
+
+\begin{figure}
+  \includegraphics{figs/bound.pdf}
+
+  \caption{
+    The line bounding $\delta$ in the complex plane as a function of
+    $\Delta=1,2,\ldots,6$ (inner to outer). Notice that for $\Delta\leq4$,
+    $|\delta|=\Delta$ is saturated for positive real $\delta$, but is not for
+    $\Delta>4$, and $\Delta=4$ has a cusp in the boundary. This is due to
+    $\Delta=4$ corresponding to the maximum distance between any two points on
+    the real sphere.
+  }
+\end{figure}
 
 \section{The $p$-spin spherical models: numerics}
 
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