  At the mean-field level, on fully connected lattices, several disordered spin
models have been shown to belong to the universality class of "structural
glasses", with a "random first-order transition" (RFOT) characterized by a
discontinuous jump of the order parameter and no latent heat. However, their
behavior in finite dimensions is often drastically different, displaying either
no glassiness at all or a conventional spin-glass transition. We clarify the
physical reasons for this phenomenon and stress the unusual fragility of the
RFOT to short-range fluctuations, associated e.g. with the mere existence of a
finite number of neighbors. Accordingly, the solution of fully connected models
is only predictive in very high dimension whereas, despite being also
mean-field in character, the Bethe approximation provides valuable information
on the behavior of finite-dimensional systems. We suggest that before embarking
on a full-blown account of fluctuations on all scales through computer
simulation or renormalization-group approach, models for structural glasses
should first be tested for the effect of short-range fluctuations and we
discuss ways to do it. Our results indicate that disordered spin models do not
appear to pass the test and are therefore questionable models for investigating
the glass transition in three dimensions. This also highlights how nontrivial
is the first step of deriving an effective theory for the RFOT phenomenology
from a rigorous integration over the short-range fluctuations.
