On the behavior of quantum walks confined to a cycle coupled with a half line

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.