  The spatial coverage produced by a single discrete-time random walk, with
asymmetric jump probability $p\neq 1/2$ and non-uniform 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 $p\to
p_c$, 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.
