Fragment Formation in Biased Random Walks

TL;DR

This paper investigates the formation of fragments in a biased one-dimensional random walk, deriving exact expressions and asymptotic behavior for the probability of fragment sizes at the phase transition point.

Contribution

It provides an exact generating function for fragment probabilities and proves their asymptotic decay, advancing understanding of phase transitions in biased random walks.

Findings

Derived an exact generating function for fragment size distribution at criticality

Proved asymptotic decay of fragment probability as 3/[l(log l)^2]

Identified phase transition from a single cluster to multiple fragments

Abstract

We analyse a biased random walk on a 1D lattice with unequal step lengths. Such a walk was recently shown to undergo a phase transition from a state containing a single connected cluster of visited sites to one with several clusters of visited sites (fragments) separated by unvisited sites at a critical probability p_c, [PRL 99, 180602 (2007)]. The behaviour of rho(l), the probability of formation of fragments of length l is analysed. An exact expression for the generating function of rho(l) at the critical point is derived. We prove that the asymptotic behaviour is of the form rho(l) ~ 3/[l(log l)^2].