It is well known that in dimension less than three a random walk is recurrent if and only if it has no drift. In higher dimensions this is no longer true. In this talk we will consider a class of ℤ^d extensions of Markov maps endowed with natural probability measures as generalisations of random walks. In this setting an appropriate replacement for the classical recurrence formula is to consider recurrent sets and ask for their Hausdorff dimensions. We will show that in our class of ℤ^d extensions, analogous to random walks on ℤ and ℤ², the following equivalence holds: the recurrent set is not of full dimension if and only if the system exhibits drift with respect to the natural measure. These results generalise previous work by Maik Gröger, Johannes Jaerisch and Marc Kesseböhmer.