  We consider reflecting random walks on the nonnegative integers with drift of
order 1/x at height x. We establish explicit asymptotics for various
probabilities associated to such walks, including the distribution of the
hitting time of 0 and first return time to 0, and the probability of being at a
given height k at time n (uniformly in a large range of k.) In particular, for
drift of form -\delta/2x + o(1/x) with \delta > -1, we show that the
probability of a first return to 0 at time n is asymptotically n^{-c}\phi(n),
where c = (3+\delta)/2 and \phi is a slowly varying function given explicitly
in terms of the o(1/x) terms.
