Critical scaling in standard biased random walks

TL;DR

This paper investigates the critical phenomena in biased random walks on a 1D lattice, revealing scale-free distributions and fractal structures near a critical bias point, combining analytical and simulation approaches.

Contribution

It identifies and characterizes a critical point in biased random walks, showing scale-free fragment distributions and fractal crack patterns, which were not previously detailed.

Findings

At the critical bias, fragment length distribution becomes scale-free.

The spatial distribution of unvisited sites forms a fractal set.

The model exhibits rich critical behavior with nontrivial exponents.

Abstract

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.