Erosion by a one-dimensional random walk

Abstract

We consider a model introduced by Baker et al. [Phys. Rev. E 88, 042113 (2013)PLEEE81539-375510.1103/PhysRevE.88.042113] of a single lattice random walker moving on a domain of allowed sites, surrounded by blocked sites. The walker enlarges the allowed domain by eroding the boundary at its random encounters with blocked boundary sites: attempts to step onto blocked sites succeed with a given probability and convert these sites to allowed sites. The model interpolates continuously between the Pólya random walker on the one-dimensional lattice and a "blind" walker who attempts freely, but always aborts, moves to blocked sites. We obtain some exact results about the walker's location and the ra..