From 95dac43e6bb643e1cb64167671627f277208d4d1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 8 Jul 2022 12:33:30 +0200
Subject: More pedogogical introduction to the model.

---
 frsb_kac-rice.tex | 10 +++++++---
 1 file changed, 7 insertions(+), 3 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 70c61f2..a22b7b2 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -69,9 +69,13 @@ For definiteness, we consider the mixed $p$-spin spherical model, whose Hamilton
 is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the sphere
 $\|\mathbf s\|^2=N$.  The coupling coefficients $J$ are taken at random, with
 zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that
-the energy is typically extensive. This implies that the covariance of the
-energy with itself depends only on the dot product (or overlap) between two
-configurations. In particular, one has
+the energy is typically extensive. The factors $a_p$ in the variances are
+freely chosen constants that define the particular model. For instance, the
+so-called `pure' models have $a_p=1$ for some $p$ and all others zero.
+
+The variance of the couplings implies that the covariance of the energy with
+itself depends only on the dot product (or overlap) between two configurations.
+In particular, one has
 \begin{equation}
   \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right)
 \end{equation}
-- 
cgit v1.2.3-70-g09d2

