  We study the role of fluctuations on the thermodynamic glassy properties of
plaquette spin models, more specifically on the transition involving an overlap
order parameter in the presence of an attractive coupling between different
replicas of the system. We consider both short-range fluctuations associated
with the local environment on Bethe lattices and long-range fluctuations that
distinguish Euclidean from Bethe lattices with the same local environment. We
find that the phase diagram in the temperature-coupling plane is very sensitive
to the former but, at least for the $3$-dimensional (square pyramid) model,
appears qualitatively or semi-quantitatively unchanged by the latter. This
surprising result suggests that the mean-field theory of glasses provides a
reasonable account of the glassy thermodynamics of models otherwise described
in terms of the kinetically constrained motion of localized defects and taken
as a paradigm for the theory of dynamic facilitation. We discuss the possible
implications for the dynamical behavior.

Statistical Mechanics;Disordered Systems and Neural Networks