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| -rw-r--r-- | frsb_kac-rice.tex | 26 |
1 files changed, 22 insertions, 4 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index ace1b28..7bf0c66 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -78,7 +78,25 @@ minima of two total energies such that one of the systems is frozen have nonzero \section{The model} -Here we consider, for definiteness, the `toy' model introduced by M\'ezard and Parisi +Here we consider, for definiteness, the `toy' model introduced by M\'ezard and Parisi (????? what) + +The mixed $p$-spin model +\begin{equation} + H(s)=\sum_p\frac{a_p^{1/2}}{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} +\end{equation} +for $\overline{J^2}=p!/2N^{p-1}$. Then +\begin{equation} + \overline{H(s_1)H(s_2)}=Nf\left(\frac{s_1\cdot s_2}N\right) +\end{equation} +for +\begin{equation} + f(q)=\frac12\sum_pa_pq^p +\end{equation} +Can be thought of as a model of generic gaussian functions on the sphere. +To constrain the model to the sphere, we use a Lagrange multiplier $\mu$, with the total energy being +\begin{equation} + H(s)+\frac\mu2(N-s\cdot s) +\end{equation} @@ -267,21 +285,21 @@ when $\mu=\mu_m$, the critical points are marginal minima. e^{nN\mathcal D(\mu)} \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a} \exp\left[ - \sum_{ab}^n + \frac12\sum_{ab}^n (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)\overline{H(s_a)H(s_b)} \right] \\ &=\frac1N\lim_{n\to0}\frac\partial{\partial n} e^{nN\mathcal D(\mu)} \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a} \exp\left[ - N\sum_{ab}^n + \frac N2\sum_{ab}^n (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)f(s_as_b/N) \right] \\ &=\frac1N\lim_{n\to0}\frac\partial{\partial n} e^{nN\mathcal D(\mu)} \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a} \exp\left[ - N\sum_{ab}^n + \frac N2\sum_{ab}^n ( \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\frac{\hat s_as_b}Nf'(s_as_b/N)+\frac{\hat s_a\hat s_b}Nf'(s_as_b/N) +\left(\frac{\hat s_as_b}N\right)^2f''(s_as_b/N) |