We consider a two-type (red and blue or R and B) particle population that evolves on the d-dimensional lattice according to some reaction-diffusion process R+B→2R and starts with a single red particle and a density ρ of blue particles. For two classes of models we give an upper bound on the propagation velocity of the red particles front with explicit dependence on ρ.
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In the first class of models red particles evolve with a diffusion constant DR=1. Blue particles evolve with a possibly time-dependent jump rate DB≥0, or, more generally, follow independent copies of some bistochastic process. Examples of bistochastic process also include long-range random walks with drift and various deterministic processes. For this class of models we get in all dimensions an upper bound of order $\max(\rho,\sqrt{\rho})$ that depends only on ρ and d and not on the specific process followed by blue particles, in particular that does not depend on DB. We argue that for d≥2 or ρ≥1 this bound can be optimal (in ρ), while for the simplest case with d=1 and ρ<1 known as the frog model, we give a better bound of order ρ.
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In the second class of models particles evolve according to Kawasaki dynamics, that is, with exclusion and possibly attraction, inside a large two-dimensional box with periodic boundary conditions (this turns into simple exclusion when the attraction is set to zero). In a low density regime we then get an upper bound of order $\sqrt{\rho}$ . This proves a long-range decorrelation of dynamical events in this low density regime.