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 $R+B\to 2R$ and starts with a single red particle and a density $\rho$ 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 $\rho$. In the first class of models red and blue particles respectively evolve with a diffusion constant $D_R=1$ and a possibly time dependent jump rate $D_B \geq 0$ -- more generally blue particles follow some independent bistochastic process and this also includes long range random walks with drift and various deterministic processes. We then get in all dimensions an upper bound of order $\max(\rho,\sqrt\rho)$ that depends only on $\rho$ and $d$ and not on the specific process followed by blue particles, in particular that does not depend on $D_B$. We argue that for $d \geq 2$ or $\rho \geq 1$ this bound can be optimal (in $\rho$), while for the simplest case with $d=1$ and $\rho < 1$ known as the frog model, we give a better bound of order $\rho$. In the second class of models particles evolve with exclusion and possibly attraction inside a large two-dimensional box with periodic boundary conditions according to Kawasaki dynamics (that 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.