  The most recent manifestation of cold Rydberg atom quantum simulators that
employs tailored optical tweezer arrays enables the study of many-body dynamics
under so-called facilitation conditions. We show how the facilitation mechanism
yields a Hilbert space structure in which the many-body states organize into
synthetic lattices, which feature in general one or several flat bands and may
support immobile localized states. We focus our discussion on the case of a
ladder lattice geometry for which we analyze in particular the influence of
disorder generated by the uncertainty of the atomic positions. The localization
properties of this system are characterized through two localization lengths
which are found to display anomalous scaling behavior at certain energies.
Moreover, we discuss the experimental preparation of an immobile localized
state, and analyze disorder-induced propagation effects.
