Caging and mosaic length scales in plaquette spin models of glasses

We consider two systems of Ising spins with plaquette interactions. They are simple models of glasses which have dual representations as kinetically constrained systems. These models allow an explicit analysis using the mosaic, or entropic droplet, approach of the random first-order transition theory of the glass transition. We show that the low-temperature states of these systems resemble glassy mosaic states, despite the fact that excitations are localized and that there are no static singularities. By means of finite-size thermodynamics we study a generalized caging effect whereby the system is frozen on short length scales, but free at larger length scales. We find that the freezing length scales obtained from statics coincide with those relevant to dynamic correlations, as expected in the mosaic view. The simple nucleation arguments of the mosaic approach, however, do not give the correct relation between freezing lengths and relaxation times, as they do not capture the transition states for relaxation. We discuss how these results make a connection between the mosaic and the dynamic facilitation views of glass formers.