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Diffstat (limited to 'frsb_kac-rice.tex')
| -rw-r--r-- | frsb_kac-rice.tex | 7 |
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} |