Storing Infinite Numbers of Patterns in a Spin-Glass Model of Neural Networks

The Hopfield model for a neural network is studied in the limit when the number $p$ of stored patterns increases with the size $N$ of the network, as $p=\ensuremath{\alpha}N$. It is shown that, despite its spin-glass features, the model exhibits associative memory for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, ${\ensuremath{\alpha}}_{c}\ensuremath{\gtrsim}0.14$. This is a result of the existence at low temperature of $2p$ dynamically stable degenerate states, each of which is almost fully correlated with one of the patterns. These states become ground states at $\ensuremath{\alpha}<0.05$. The phase diagram of this rich spin-glass is described.