Fragment Formation in Biased Random Walks

Details Fragment Formation in Biased Random Walks Fragment Formation in Biased Random Walks

2008-08-31

arxiv.org/abs/0809.0147v1

J. Stat. Mech. (2008) P10002

Physics

Condensed Matter

Statistical Mechanics

6 pages, 2 figures

Scientific paper

10.1088/1742-5468/2008/10/P10002

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].

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