THERMODYNAMIC LIMIT OF THE Q-STATE POTTS-HOPFIELD MODEL WITH INFINITELY MANY PATTERNS

We prove the almost sure convergence of the free energy and of the overlap order parameters in a q-state version of the Hopfield neural network model. We compute explicitly these limits for all temperatures different from some critical value. The number of stored patterns is allowed to grow with the size of the system N like (alpha/ln q) ln N. We study the limiting behavior of the extremal states of the model that are the measures induced on the Gibbs measures by the overlap parameters.