  We study a random walk in random environment on the non-negative integers.
The random environment is not homogeneous in law, but is a mixture of two kinds
of site, one in asymptotically vanishing proportion. The two kinds of site are
(i) points endowed with probabilities drawn from a symmetric distribution with
heavy tails at 0 and 1, and (ii) `fast points' with a fixed systematic drift.
Without these fast points, the model is related to the diffusion in
heavy-tailed (`stable') random potential studied by Schumacher and Singh; the
fast points perturb that model. The two components compete to determine the
behaviour of the random walk; we identify phase transitions in terms of the
model parameters. We give conditions for recurrence and transience, and prove
almost-sure bounds for the trajectories of the walk.
