DIRECTED LINES IN SPARSE POTENTIALS

We present a continuum formulation of a (d+1)-dimensional directed lin e interacting with sparse potentials (i.e., d-dimensional potentials d efined 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 probabilit y density P(x,t) of the directed line. The effects of these discontinu ities are studied in detail for the simple case of a single defect. We then investigate sparse columnar potentials defined as a periodic arr ay of defects in (2+1) dimensions, and solve exactly for P. A nontrivi al binding-unbinding transition is found.