Unlike finite-state Markov chains, interacting particle systems on infinite graphs may admit multiple stationary measures, and even convergence to one of these is not guaranteed. One particular obstruction is the existence of time-periodic orbits, where the system returns to its initial distribution after a fixed period without being stationary, thereby spontaneously breaking the continuous time-translation symmetry. The conditions under which such noisy systems can maintain sufficient spatial order to sustain such time-periodic behaviour have been a contentious issue in the physics literature and a mathematical treatment of such broken-symmetry, macroscopically time-dependent states has been mostly limited to mean-field like approximations.
In this talk, we present some recent progress on two questions: (a) in which spatial dimensions d is such time-periodic behaviour possible? (b) What quantitative properties, such as the rate of decay of truncated correlations, can we infer from the existence of a time-periodic orbit?