95-517 G.R. Guerberoff, and G.A. Raggio

ON THE FREE ENERGY OF THE HOPFIELD MODEL (107K, LaTeX) Dec 1, 95

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Abstract. The general theory of inhomogeneous mean-field systems of Ref. \cite{RaWe2} provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model \[ H_{N,p}^{\{ \mbox{\boldmath $\xi$} \}} (S) = - \frac{1}{2N} \sum_{i,j =1}^N \sum_{\mu =1}^p \xi^{\mu}_i \xi^{\mu}_j S_i S_j \] for Ising spins $S_i$ and $p$ random patterns $\mbox{\boldmath $\xi$}^{\mu} = ( \xi^{\mu}_1, \xi^{\mu}_2, \cdots , \xi^{\mu}_N )$ under the assumption that \[ \lim_{N \to \infty} N^{-1} \sum_{i=1}^N \delta_{\mbox{\boldmath $\xi$}_i} = \lambda \; \; \; , \; \; \; \mbox{\boldmath $\xi$}_i = ( \xi^1_i, \xi^2_i, \cdots , \xi^p_i )\] exists (almost surely) in the space of probability measure over $p$ copies of $\{ -1 , 1 \}$. Including an ``external field'' term $ - \sum_{\mu=1}^p h^{\mu} \sum_{i=1}^N \xi^{\mu}_i S_i$, we give a number of general properties of the free-energy density and compute it for a): $p=2$ in general, and b): $p$ arbitrary when $\lambda$ is uniform and at most the two components $ h^{\mu_1}$ and $h^{\mu_2}$ are non-zero, obtaining the (almost sure) formula \[ f( \beta , {\bf h}) = \frac{1}{2} f^{cw}( \beta , h^{\mu_1}+h^{\mu_2} ) + \frac{1}{2} f^{cw} ( \beta , h^{\mu_1}-h^{\mu_2}) \] for the free energy, where $f^{cw}$ denotes the limiting free energy density of the Curie-Wei{\ss} model with unit interaction constant. In both cases, we obtain explicit formulae for the limiting (almost sure) values of the so-called overlap parameters \[ m_N^{\mu} ( \beta , {\bf h } ) = N^{-1} \sum_{i=1}^N \xi^{\mu}_i < S_i > \] in terms of the Curie-Wei{\ss} magnetizations. For the general i.i.d. case with $Prob \{ \xi^{\mu}_i= \pm 1 \} = (1/2) \pm \epsilon$, we obtain the lower bound $1+ 4 \epsilon^2 (p-1)$ for the temperature $T_c$ separating the trivial free regime where the overlap vector is zero, from the non-trivial regime where it is non-zero. This lower bound is exact for: $p=2$, or $\epsilon =0$, or $\epsilon = \pm 1/2$. For $p=2$ we identify an intermediate temperature region between $T_*= 1- 4 \epsilon^2$ and $T_c = 1 + 4 \epsilon^2$ where the overlap vector is homogeneous (i.e. all its components are equal) and non-zero. $T_*$ marks the transition to the non-homogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous non-zero regime exists for $p \geq 3$ and that $T_* = \max \{ 1- 4 \epsilon^2 (p-1), 0 \}$.

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