Random walk in dynamic random environment with long-range space correlations

Models of random walks (RW) in dynamic random environment (RE) are usually considered under some space-time mixing conditions with sufficient decay. We study a discrete-time model on $\mathbb Z$ in an environment independent in time, but with non-absolutely summable space correlations. We show that an a.-s. quenched Central Limit Theorem (CLT) holds, with the same leading term as in the uncorrelated case, and the same order of decay of the first correction. Some conclusions are drawn on the type of correlations that could modify the leading terms of the CLT asymptotics.