Fluctuations of the front in a one dimensional model for the spread of an infection
Résumé
We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent continuous-time nearest-neighbor symmetric random walks on the integer lattice $\Z$ with jump rates $D_R$ for red particles and $D_B$ for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the front, defined as the rightmost position of a red particle. For the case $D_R=D_B$ (in fact, for a general $d-$dimensional version of it), Kesten and Sidoravicius established that the front moves ballistically, and more precisely that it satisfies a law of large numbers. In this paper, we prove that a central limit theorem for the front holds when $D_R=D_B$. Moreover, this result can be extended to the case where $D_R>D_B$, up to modifying the dynamics so that blue particles turn red upon contact with a site that has previously been occupied by a red particle. Our approach is based on the definition of a renewal structure, extending ideas developed by Comets, Quastel and Ramí rez for the so-called frog model, where $D_B=0$.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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