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
pc
(Anteneodo and Morgado 2007 Phys. Rev. Lett. 99 180602). The behaviour of
ρ(l), the probability of formation of fragments of length
l, is analysed. An exact expression for the generating function of
ρ(l)
at the critical point is derived. We prove that the asymptotic behaviour is of the form 
.