  We analyze a semi-infinite one-dimensional random walk process with a biased
motion that is incremental in one direction and long-range in the other. On a
network with a fixed hierarchy of long-range jumps, we find with exact
renormalization group calculations that there is a dynamical transition between
a localized adsorption phase and an anomalous diffusion phase in which the
mean-square displacement exponent depends non-universally on the Bernoulli
coin. We relate these results to similar findings of unconventional phase
behavior in hierarchical networks.
