Random walks in a~random (fluctuating) environment

C. Boldrighini ; R. A. Minlos ; A. Pellegrinotti

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 62 (2007) no. 4, pp. 663-712

Résumé

The main purpose of this paper is to prove the central limit theorem for the position at large times of a particle performing a discrete-time random walk on the lattice $\mathbb Z^d$ when the particle interacts with a random ‘environment’ (and starts out at a fixed point of the lattice). Two cases are considered for the distribution of the particle position: first, the distribution when the configuration of the ‘environment’ (that is, of the random field) is fixed at all points of the ‘space-time’ $\mathbb Z^{d+1}$ (the so-called quenched model), and, second, the distribution induced by the joint evolution of the environment and the particle position under the assumption that the pair forms a Markov chain (the annealed model). Two cases are considered also for the quenched model: the values of the field at all points of ‘space-time’ are independent and identically distributed, or the values of the field at different times are linked by a homogeneous Markov chain. In the case of quenched models the central limit theorem with one and the same limit law is true for almost all configurations of the ‘environment’, and in the case of annealed models it is true for any initial distribution of the field. Besides the central limit theorem, the paper briefly treats some other topics related to these models (decay of correlations, large deviations, ‘the field from the viewpoint of a particle’, and so on).