Critical scaling in standard biased random walks

Details Critical scaling in standard biased random walks Critical scaling in standard biased random walks

2008-01-25

arxiv.org/abs/0801.4005v1

Phys. Rev. Lett. 99, 180602 (2007)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 4 figures

Scientific paper

10.1103/PhysRevLett.99.180602

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.

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