Scaling of a random walk on a supercritical contact process
نویسندگان
چکیده
A proof is provided of 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 is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For 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. The qualitative dependence of the speed and the volatility on the infection parameter is investigated, and some open problems are mentioned. Acknowledgment. The authors would like to thank Stein Bethuelsen and Markus Heydenreich for stimulating discussions, and the anonymous referee for useful suggestions. MSC 2000. Primary 60F15, 60K35, 60K37; Secondary 82B41, 82C22, 82C44.
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