  When confined to a topological environment consisting of a cycle coupled with
a half-line, quantum walks exhibit long-term statistical tendencies which
differ dramatically from the tendencies of classical random walks in the same
environment. In particular, as suggested by numerical simulations, the
probability distribution of the walker's position resolves, in part, into a
non-vanishing distribution on the cycle and, in part, into a ballistic
distribution on the half-line. By contrast, for a classical random walk, the
probability distribution of the walker's position tends always to vanish on the
cycle and to migrate completely to the half-line as a purely diffusive process.
