In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. In this paper, we are interested in a class of weakly inhomogeneous random walks termed Random Walk with Random Transition Probabilities — we refer to [18] for the terminology. As an application, we give a criterion for the recurrence or transience of these processes in the discrete Abelian case. This criterion is deduced using Fourier analysis of Markov additive processes and a perturbation argument of a Markov operator. The latter extends the results of the literature since it does not involve a quasi-compacity condition on the operator. Finally, we apply this criterion on some well known examples of random walks on directed graphs embedded in Z 2. Despite the type problem has been already solved for these examples, our analysis brought a new insight to this problematic.