  In this paper we introduce and exploit the real replica approach for a
minimal generalization of the Hopfield model, by assuming the learned patterns
to be distributed accordingly to a standard unit Gaussian. We consider the high
storage case, when the number of patterns is linearly diverging with the number
of neurons. We study the infinite volume behavior of the normalized momenta of
the partition function. We find a region in the parameter space where the free
energy density in the infinite volume limit is self-averaging around its
annealed approximation, as well as the entropy and the internal energy density.
Moreover, we evaluate the corrections to their extensive counterparts with
respect to their annealed expressions. The fluctuations of properly introduced
overlaps, which act as order parameters, are also discussed.
