INTERACTING RANDOM-WALK IN A DYNAMICAL RANDOM ENVIRONMENT .1. DECAY OF CORRELATIONS

We consider a random walk X(t), t is-an-element-of Z+ and a dynamical random field xi(t) (x), x is-an-element-of Z(nu) (t is-an-element-of Z +) in mutual interaction with each other. The interaction is small, an d the model is a perturbation of an unperturbed model in which walk an d field evolve independently, the walk according to i.i.d. finite rang e jumps, and the field independently at each site x is-an-element-of Z (nu), according to an ergodic Markov chain. Our main result in Part I concerns the asymptotics of temporal correlations of the random field, as seen in a fixed frame of reference. We prove that it has a ''long time tail'' falling off as an inverse power of t. In Part II we obtain results on temporal correlation in a frame of reference moving with t he walk.