Weak dependence for a class of local functionals of Markov chains on ${\mathbb{Z}}^d$

In some papers on infinite Markov chains in ${\mathbb{Z}}^d$, and notably in the work of R.A. Minlos and collaborators, one can prove the existence of a spectral gap for a suitable subspace of local functions. We consider functions of the type $f(\widehat{\eta })$, where $\widehat \eta= {\eta_{t}}_{t=0}^{\infty}$isthesequenceofthestates,and$f$ is local. In the case of a simple example of random walk in random environment with mutual interaction we show that there is a natural class of functions $f$, related to the H"older continuos functions ${\mathcal{C}}^{\alpha }$on the torus $T^1$, with $\alpha \in (0,1)$ large enough, depending on the spectral gap, for which the Central Limit Theorem holds for the sequence $f(S^{k}\widehat \eta)$,$k=0,1,\ldots$,where$S$ is the time shift.