From 7f42464134e8291548be5f0727fb62033192d20f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 24 Aug 2023 12:28:51 +0200
Subject: Added sentence clarifying that H is centered.

---
 when_annealed.tex | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index b07e3cd..5dc4065 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -110,6 +110,7 @@ product (or overlap) between the two configurations:
 \begin{equation} \label{eq:covariance}
   \overline{H(\pmb\sigma_1)H(\pmb\sigma_2)}=\frac1Nf\bigg(\frac{\pmb\sigma_1\cdot\pmb\sigma_2}N\bigg)
 \end{equation}
+We will further take the distribution of $H$ to be centered, i.e., $\overline{H(\pmb\sigma)}=0$ for all $\pmb\sigma\in S^{N-1}$, which is equivalent to the absence of any deterministic term (or spike) in the function.
 Specifying the covariance function $f$ uniquely specifies the model. The series
 coefficients of $f$ need to be nonnnegative in order for $f$ to be a
 well-defined covariance. The case where $f$ is a homogeneous polynomial has
@@ -130,7 +131,7 @@ called $3+s$ models.\footnote{
   trivial overlap $q_0$ is also important in situations where a deterministic
   field (or spike) is present, as in \cite{Ros_2019_Complex}, but deterministic
   fields are likewise not considered here.
-}These are examples of \emph{mixed} spherical models, which have been studied
+} These are examples of \emph{mixed} spherical models, which have been studied
 in the physics and statistics literature and host a zoo of complex orders and
 phase transitions \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical,
   Krakoviack_2007_Comment, Crisanti_2007_Amorphous-amorphous,
-- 
cgit v1.2.3-54-g00ecf

