  We study, in d-dimensions, the random walker with geometrically shrinking
step sizes at each hop. We emphasize the integrated quantities such as
expectation values, cumulants and moments rather than a direct study of the
probability distribution. We develop a 1/d expansion technique and study
various correlations of the first step to the position as ti me goes to
infinity. We also show and substantiate with a study of the cumulants that to
order 1/d the system admits a continuum counterpart equation which can be
obtained with a generalization of the ordinary technique to obtain the
continuum limit. We also advocate that this continuum counterpart equation,
which is nothing but the ordinary diffusion equation with a diffusion constant
decaying exponentially in continuous time, captures all the qualitative aspects
of t he discrete system and is often a good starting point for quantitative
approximations.
