Critical scaling in standard biased random walks

The spatial coverage produced by a single discrete-time random walk, with an asymmetric jump probability p not equal 1/2 and nonuniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented with Monte Carlo simulations. We show that, for appropriate step sizes, the model displays a critical phenomenon, at p=p(c). Its scaling properties as well as the main features of the fragmented coverage occurring in the vicinity of the critical point are shown. In particular, in the limit p -> p(c), the distribution of fragment lengths is scale-free, with nontrivial exponents. Moreover, the spatial distribution of cracks (unvisited sites) defines a fractal set over the spanned interval. Thus, from the perspective of the covered territory, a very rich critical phenomenology is revealed in a simple one-dimensional standard model.