Directed lines in sparse potentials

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.