An upper bound for front propagation velocities inside moving populations

Abstract

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. 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. 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.