  The relativistic Hopfield model constitutes a generalization of the standard
Hopfield model that is derived by the formal analogy between the
statistical-mechanic framework embedding neural networks and the Lagrangian
mechanics describing a fictitious single-particle motion in the space of the
tuneable parameters of the network itself. In this analogy the cost-function of
the Hopfield model plays as the standard kinetic-energy term and its related
Mattis overlap (naturally bounded by one) plays as the velocity. The
Hamiltonian of the relativisitc model, once Taylor-expanded, results in a
P-spin series with alternate signs: the attractive contributions enhance the
information-storage capabilities of the network, while the repulsive
contributions allow for an easier unlearning of spurious states, conferring
overall more robustness to the system as a whole. Here we do not deepen the
information processing skills of this generalized Hopfield network, rather we
focus on its statistical mechanical foundation. In particular, relying on
Guerra's interpolation techniques, we prove the existence of the infinite
volume limit for the model free-energy and we give its explicit expression in
terms of the Mattis overlaps. By extremizing the free energy over the latter we
get the generalized self-consistent equations for these overlaps, as well as a
picture of criticality that is further corroborated by a fluctuation analysis.
These findings are in full agreement with the available previous results.
