  We present a continuum formulation of a (d+1)-dimensional directed line
interacting with sparse potentials (i.e. d-dimensional potentials defined only
at discrete longitudinal locations.) An iterative solution for the partition
function is derived. The impulsive influence of the potentials induces
discontinuities in the evolution of the probability density P(x,t) of the
directed line. The effects of these discontinuities are studied in detail for
the simple case of a single defect. We then investigate sparse columnar
potentials defined as a periodic array of defects in (2+1) dimensions, and
solve exactly for P. A non-trivial binding/unbinding transition is found.
