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-rw-r--r--frsb_kac-rice.tex7
1 files changed, 5 insertions, 2 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 3b17093..c5a288b 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -60,8 +60,11 @@ Here we consider, for definiteness, the mixed $p$-spin model, whose Hamiltonian
\begin{equation}
H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
\end{equation}
-is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the sphere $\|\mathbf s\|^2=N$.
-The coupling coefficients are taken at random, with zero mean and covariance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$. This implies that the covariance of the energy with itself depends only on the dot product, or overlap, between two configurations, and in particular that
+is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the sphere
+$\|\mathbf s\|^2=N$. The coupling coefficients are taken at random, with zero
+mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$. This implies that
+the covariance of the energy with itself depends only on the dot product, or
+overlap, between two configurations, and in particular that
\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}