  In this work we develop analytical techniques to investigate a broad class of
associative neural networks set in the high-storage regime. These techniques
translate the original statistical-mechanical problem into an
analytical-mechanical one which implies solving a set of partial differential
equations, rather than tackling the canonical probabilistic route. We test the
method on the classical Hopfield model - where the cost function includes only
two-body interactions (i.e., quadratic terms) - and on the "relativistic"
Hopfield model - where the (expansion of the) cost function includes p-body
(i.e., of degree p) contributions. Under the replica symmetric assumption, we
paint the phase diagrams of these models by obtaining the explicit expression
of their free energy as a function of the model parameters (i.e., noise level
and memory storage). Further, since for non-pairwise models ergodicity breaking
is non necessarily a critical phenomenon, we develop a fluctuation analysis and
find that criticality is preserved in the relativistic model.
