Fluctuations of the front in a one-dimensional model for the spread of an infection

• DOI: 10.1214/15-aop1034
• Publication Date: 2016-08-02T11:59:15Z
• Authors: Jean Bérard; Alejandro Ramírez
• Keywords: Interacting particle system

We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent nearest-neighbor continuous-time symmetric random walks on integer lattice $\mathbb{Z}$ with jump rates $D_{R}$ for $D_{B}$ particles, interaction rule being that turn upon contact a particle. The initial condition consists i.i.d. Poisson particle numbers at each site, left origin red, while right are blue. interested in dynamics front, defined as rightmost position For case $D_{R}=D_{B}$, Kesten Sidoravicius established front moves ballistically, more precisely it satisfies law large numbers. Their proof is based multi-scale renormalization technique, combined approximate sub-additivity arguments. In this paper, we build renewal structure propagation process, corollary obtain central limit theorem when $D_{R}=D_{B}$. Moreover, result can be extended to where $D_{R}>D_{B}$, up modifying so site has previously been occupied by Our approach extends developed Comets, Quastel Ramírez so-called frog model, which corresponds $D_{B}=0$ case.