Scaling of a random walk on a supercritical contact process

den Hollander, F.; dos Santos, R. S.

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1276-1300.

Nous prouvons une loi forte des grands nombres pour une marche aléatoire dans un milieu aléatoire dynamique donné par un processus de contact sur-critique unidimensionnel en équilibre. La preuve utilise un argument de couplage basé sur l'observation que la marche est finalement confinée dans l'union de cônes spatio-temporels inclus dans les clusters d'infection générés par des infections individuelles. Si les taux locaux de saut de la marche sont plus petits que la vitesse de propagation de l'infection, la marche est finalement confinée dans un seul cône, ce qui entraîne l'existence de temps de régénération en lesquels la marche oublie son passé. Ces temps de régénération sont utilisés pour prouver un théorème central limite fonctionnel et un principe de grandes déviations sous la loi “annealed.” La dépendance de la vitesse et de la variance asymptotiques par rapport au paramètre d'infection est étudiée, et quelques problèmes ouverts sont mentionnés.

We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space-time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle under the annealed law. The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.

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