We show that the dynamics of simple disordered models, like the directed Trap Model and the Random Energy Model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in
these models to the extreme value statistics of the associated random energy landscape.
We calculate exactly the first cumulants of the integrated current and of the activity (which is the total number of changes of configurations) of the symmetric simple exclusion process (SSEP) on a ring with periodic boundary conditions. Our results
indicate that for large system sizes the large deviation functions of the current and of the activity take a universal scaling form, with the same scaling function for both quantities. This scaling function can be understood either by an analysis of Bethe ansatz equations or in terms of a theory based on fluctuating hydrodynamics or on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim.
