DYNAMICS OF ISING RANDOM-BOND MODELS - NEURAL-NETWORK AND RANDOM-ANISOTROPY-AXIS MODEL

A discrete-time retrieval dynamics for a class of random-bond models with infinite range interactions is studied in a unified picture. The Hopfield model of neural networks with the Hebb learning rule is considered together with the Ising random-anisotropy-axis model in the strong-anisotropy limit. The main overlap (magnetization), the residual overlap and its dispersion, as well as the correlation between the Gaussian component of the residual overlap and the initial value of the latter, obtained in the first step of a recursion relation, are used to infer the structure of the recursion for large times. A crucial assumption is the strong stationarity of the Gaussian component. The dynamics are discussed for finite alpha = p/N (the storage ratio in the neural network problem or the ratio of random-axis components per site) in the limit where both p and N go to infinity. The long-time behaviour of the theory is shown to yield the equilibrium solution of an earlier work in mean-field theory, for a tri-modal distribution of random-axis components. Explicit results for the basins of attraction of either a ferromagnetic or a spin-grass phase are obtained, as well as the relaxation time with a square-root power-law decay near saturation.