Abstract. Inspired by investigations of the geometry of limit sets of geometrically infinite horospherically tame Kleinian groups, we study a certain Cantor-like set which represents a 1-dimensional model of the conformal dynamics inside a tame singly cusped parabolic ending. Our main result is to show that the set of non-recurrent points within this set has full Hausdorff dimension. The proof is based on the transience of a non-symmetric Cauchy-type random walk.