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| -rw-r--r-- | frsb_kac-rice.tex | 10 |
1 files changed, 7 insertions, 3 deletions
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} |