Random walks in space time mixing environments

We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure. 1 The results Random walks in random environments are walks where the transition probabilities are themselves random variables (see [22, 23] for recent reviews of the literature). The environments can be divided into two main classes: static and dynamical ones. In the first case, the transition probabilities are given once and for all, and the walk can be “trapped” for a long time in some regions because the transition probabilities happen to favour motion towards that region. This may lead to anomalously slow diffusion in one dimension, as was shown by Sinai [20]. In [2, 21], it is shown that, in three or more dimensions and for weak disorder (almost deterministic walks), ordinary diffusion takes place. In dynamical environments, the random transition probabilities change with time and trapping does not occur, so that one expects ordinary diffusion to hold in all dimensions. Although simpler than the static environments, the dynamical ones are not trivial to analyze; see [11] for recent and general results and for references to earlier ones. We consider in this paper a rather general class of space-time mixing environments. This means that the transition probabilities at different times and spatial points are weakly correlated and moreover the randomness is weak. For such environments we prove that the walks are diffusive, almost surely in the environment measure. In particular we do not assume a Markovian structure of the environment. We only assume that certain cumulants (or connected correlation functions) decay in a way that is typical of what happens in high temperature or weakly coupled Gibbs states. Partially supported by the Belgian IAP program P6/02. Partially supported by the Academy of Finland. 1 Our motivation to study this class of models comes from the consideration of random walks in a deterministic, but “chaotic” environment [12]. As shown first by Bunimovich and Sinai, the invariant measures of suitably coupled hyperbolic dynamical systems correspond, via an extension of the SRB formalism, to certain weakly coupled Gibbs states for a spin system on a space-time lattice [8, 3, 4, 5, 14, 15, 16]. A walk whose transition probabilities are local functions of such hyperbolic systems can be analyzed by the methods developed here. Random walks in such deterministic environments emerge when considering deterministic dynamics of a coupled map lattice with a global conserved quantity (”energy”). The latter in turn can be viewed as a model of coupled Hamiltonian systems where one would like to prove diffusion and Fourier’s law for heat transport. In such models the environments will have more general correlations than the Markovian ones and we expect to use the method developed in this paper. This is discussed further at the end of this Section and in [6]. The method used in the proof consists in applying a Renormalization group scheme to iterate bounds, both on the size of the coupling between the transition probabilities, and on the size of their “disorder”, i.e. of their deviation from a deterministic walk. In the long time limit, the disorder tends to zero and the resulting deterministic walk behaves diffusively. Turning to the precise models considered here, let Ω be the space of walks ω = (ω0, . . . , ωT), ωt ∈ Z, in time T and starting at ω0 = 0 and let the probability of a walk be defined as P(ω) = T−1