  Symmetry-breaking dynamical phase transitions (DPTs) abound in the
fluctuations of nonequilibrium systems. Here we show that the spectral features
of a particular class of DPTs exhibit the fingerprints of the recently
discovered time-crystal phase of matter. Using Doob's transform as a tool, we
provide a mechanism to build time-crystal generators from the rare event
statistics of some driven diffusive systems. An analysis of the Doob's smart
field in terms of the order parameter of the transition then leads to the
time-crystal exclusion process (tcEP), a stochastic lattice gas subject to an
external packing field which presents a clear-cut steady-state phase transition
to a time-crystalline phase which breaks continuous time-translation symmetry
and displays rigidity and long-range spatio-temporal order. A hydrodynamic
analysis of the tcEP transition uncovers striking similarities, but also key
differences, with the Kuramoto synchronization transition. Possible
experimental realizations of the tcEP are also discussed.
