Zero temperature parallel dynamics for infinite range spin glasses and neural networks

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Abstract
We present the results of analytical and numerical calculations for the zero temperature parallel dynamics of spin glass and neural network models. We use an analytical approach to calculate the magnetization and the overlaps after a few time steps. For the long time behaviour, the analytical approach becomes too complicated and we use numerical simulations. For the Sherrington-Kirkpatrick model, we measure the remanent magnetization and the overlaps at different times and we observe power law decays towards the infinite time limit. When one iterates two configurations in parallel, their distance d(∞) in the limit of infinite time depends on their initial distance d(0). Our numerical results suggest that d(∞) has a finite limit when d(0) → 0. This result can be regarded as a collective effect between an infinite number of spins. For the Little-Hopfield model, we compute the time evolution of the overlap with a stored pattern. We find regimes for which the system learns better after a few time steps than in the infinite time limit

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