  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].
