An upper bound for front propagation velocities inside moving populations

TL;DR

This paper establishes upper bounds on the speed of front propagation in reaction-diffusion particle systems with two types, depending on density and dimension, revealing insights into their long-range behavior.

Contribution

It provides explicit upper bounds on front velocities for two classes of models, including reaction-diffusion and exclusion processes, with bounds depending only on density and dimension.

Findings

Upper bound of order max(ρ, √ρ) for reaction-diffusion models independent of blue particles' process

Improved bound of order ρ for the 1D frog model with low density

Upper bound of order √ρ in low-density exclusion processes in 2D

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.