Random walk in dynamic environment with mutual influence

Abstract

We study a discrete time random walk in Z v in a dynamic random environment, when the evolution of the environment depends on the random walk (mutual influence). We assume that the unperturbed environment evolves independently at each site, as an ergodic Markov chain, and that the interaction is strictly local. We prove that the central limit theorem for the position X t of the random walk (particle) holds, whenever one of the following conditions is met: (i) the particle cancels the memory of the environment and the influence of the environment on the random walk is small; (ii) the exponential relaxation rate of the environment is large; (iii) the mutual interaction of the environment and the random walk is small. We also prove convergence of the distribution of the ‘environment as seen from the particle’. Proofs are obtained by cluster expansion techniques.