  We show that the dynamics of kinetically constrained models of glass formers
takes place at a first-order coexistence line between active and inactive
dynamical phases. We prove this by computing the large-deviation functions of
suitable space-time observables, such as the number of configuration changes in
a trajectory. We present analytic results for dynamic facilitated models in a
mean-field approximation, and numerical results for the Fredrickson-Andersen
model, the East model, and constrained lattice gases, in various dimensions.
This dynamical first-order transition is generic in kinetically constrained
models, and we expect it to be present in systems with fully jammed states.
