A Note on the Annealed Free Energy of the p-Spin Hopfield Model

#### H. Knopfel, M. Lowe

2007, v.13, Issue 3, 565-574

ABSTRACT

We compute the annealed free energy in the $p$-spin interaction version of the Hopfield model at high temperatures with $p\ge 3$. We show that there is a critical temperature $\tilde\beta$ depending on $p$ such that for $\beta<\sqrt{p!} \,\tilde{\beta}$ the annealed free energy of the $p$-spin Hopfield model can be computed as $(\alpha \beta^2)/ 2$. Here $\alpha =\lim_{N \to \infty} M(N)/N^{p-1}$, $M(N)$ is the number of patterns and $N$ is the number of spins. The threshold $\tilde \beta$ obeys $\lim_{p \to \infty} \tilde{\beta} = \log 2$.

Keywords: spin glasses,Hopfield model,$p$-spin models,Central Limit Theorem