Directed lines in sparse potentials

TL;DR

This paper develops a continuum model for directed lines interacting with sparse potentials, deriving an iterative solution for the partition function, and explores effects of defects and phase transitions in such systems.

Contribution

It introduces a novel continuum formulation and exact solutions for directed lines with sparse potentials, including analysis of binding transitions.

Findings

Discontinuities in probability density due to sparse potentials

Exact solution for periodic array of defects in 2+1 dimensions

Identification of a non-trivial binding/unbinding transition

Abstract

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.