  In this paper we study two non-mean-field spin models built on a hierarchical
lattice: The hierarchical Edward-Anderson model (HEA) of a spin glass, and
Dyson's hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the
existence of the thermodynamic limit of the free energy and the
replica-symmetry-breaking (RSB) free-energy bounds previously derived for the
Sherrington-Kirkpatrick model of a spin glass. These RSB mean-field bounds are
exact only if the order-parameter fluctuations (OPF) vanish: Given that such
fluctuations are not negligible in non-mean-field models, we develop a novel
strategy to tackle part of OPF in hierarchical models. The method is based on
absorbing part of OPF of a block of spins into an effective Hamiltonian of the
underlying spin blocks. We illustrate this method for DHM and show that,
compared to the mean-field bound for the free energy, it provides a tighter
non-mean-field bound, with a critical temperature closer to the exact one. To
extend this method to the HEA model, a suitable generalization of Griffith's
correlation inequalities for Ising ferromagnets is needed: Since correlation
inequalities for spin glasses are still an open topic, we leave the extension
of this method to hierarchical spin glasses as a future perspective.
