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Active-to-absorbing state phase transition in the presence of fluctuating environments: Weak and strong dynamic scaling

We investigate the scaling properties of phase transitions between survival and extinction (active-to-absorbing state phase transition, AAPT) in a model, that by itself belongs to the directed percolation (DP) universality class, interacting with a spatio-temporally fluctuating environment having its own non-trivial dynamics. We model the environment by (i) a randomly stirred fluid, governed by the Navier-Stokes (NS) equation, and (ii) a fluctuating surface, described either by the Kardar-Parisi-Zhang (KPZ) or the Edward-Wilkinson (EW) equations. We show, by using a one-loop perturbative field theoretic set up, that depending upon the spatial scaling of the variance of the external forces that drive the environment (i.e., the NS, KPZ or EW equations), the system may show {\em weak} or {\em strong dynamic scaling} at the critical point of active to absorbing state phase transitions. In the former case AAPT displays scaling belonging to the DP universality class, whereas in the latter case the universal behavior is different.