  Slow dynamics in glassy systems is often interpreted as due to thermally
activated events between "metastable" states. This emphasizes the role of
nonperturbative fluctuations, which is especially dramatic when these
fluctuations destroy a putative phase transition predicted at the mean-field
level. To gain insight into such hard problems, we consider the implementation
of a generic back-and-forth process, between microscopic theory and observable
behavior via effective theories, in a toy model that is simple enough to allow
for a thorough investigation: the one-dimensional $\varphi^4$ theory at low
temperature. We consider two ways of restricting the extent of the
fluctuations, which both lead to a nonconvex effective potential (or free
energy) : either through a finite-size system or by means of a running infrared
cutoff within the nonperturbative Renormalization Group formalism. We discuss
the physical insight one can get and the ways to treat strongly nonperturbative
fluctuations in this context.
