Varied phenomenology of models displaying dynamical large-deviation singularities

Singularities of dynamical large-deviation functions are often interpreted as the signal a phase transition and coexistence distinct phases, by analogy with correspondence between singularities free energies equilibrium behavior. Here we study models driven random walkers on lattice. These display in limit large lattice size, but extent to which each model's phenomenology resembles depends details driving. We also compare behavior ergodic non-ergodic that present singularities. argue indicate divergence model timescale, not necessarily one associated cooperative or existence phases.