From 5f9db45efcf9ffb2d97a854185a1c505d66b20d2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 Dec 2020 15:20:43 +0100
Subject: Some rearrangement.

---
 bezout.tex | 14 +++-----------
 1 file changed, 3 insertions(+), 11 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 8286208..2368a1d 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -277,8 +277,8 @@ for $\delta=\kappa a^{-(p-2)}$.
 
 With knowledge of this distribution, the integral in \eqref{eq:free.energy.a}
 may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$,
-the resulting expression is always maximized for $a=\infty$, that is, for
-unbounded complex spins. Taking this saddle gives
+the resulting expression is always maximized for $a=\infty$. Taking this saddle
+gives
 \begin{equation} \label{eq:bezout}
   \overline{\mathcal N}(\kappa,\epsilon)
   =e^{N\log(p-1)}
@@ -309,7 +309,7 @@ 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 $a=1$.  Interpreting this $a$-dependence as a cumulative count, this
+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}
@@ -321,7 +321,6 @@ points of the real pure spherical $p$-spin model. In fact, the full
 $\epsilon$-dependence of the real pure spherical $p$-spin is recovered by this
 limit as $\epsilon$ is varied.
 
-
 \begin{figure}[htpb]
   \centering
   \includegraphics{fig/complexity.pdf}
@@ -332,13 +331,6 @@ limit as $\epsilon$ is varied.
   } \label{fig:complexity}
 \end{figure}
 
-For $|\kappa|<1$,
-\begin{equation}
-  \lim_{a\to1}\overline{\mathcal N}(\kappa,\epsilon,a)
-  =0
-\end{equation}
-
-
 {\color{teal} {\bf somewhere else}
 
 Another instrument we have to study this problem is to compute the following partition function:
-- 
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