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 $\widehat \eta= \{\eta_{t}\}_{t=0}^{\infty}$ is the sequence of the states, 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.