## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 24, Number 2 (2010), 256-278.

### An upper bound for front propagation velocities inside moving populations

A. Gaudillière and F. R. Nardi

#### 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*→2*R* 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 *D*_{R}=1. Blue particles evolve with a possibly time-dependent jump rate *D*_{B}≥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 that depends only on *ρ* 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*≥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 . This proves a long-range decorrelation of dynamical events in this low density regime.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 24, Number 2 (2010), 256-278.

**Dates**

First available in Project Euclid: 20 April 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1271770271

**Digital Object Identifier**

doi:10.1214/09-BJPS030

**Mathematical Reviews number (MathSciNet)**

MR2643566

**Zentralblatt MATH identifier**

1203.60146

**Keywords**

Random walks front propagation diffusion-reaction epidemic model Kawasaki dynamics simple exclusion frog model

#### Citation

Gaudillière, A.; Nardi, F. R. An upper bound for front propagation velocities inside moving populations. Braz. J. Probab. Stat. 24 (2010), no. 2, 256--278. doi:10.1214/09-BJPS030. https://projecteuclid.org/euclid.bjps/1271770271