  Quantum walks (QWs) exhibit different properties compared with classical
random walks (RWs), most notably by linear spreading and localization. In the
meantime, random walks that replicate quantum walks, which we refer to as
quantum-walk-replicating random walks (QWRWs), have been studied in the
literature where the eventual properties of QWRW coincide with those of QWs.
However, we consider that the unique attributes of QWRWs have not been fully
utilized in the former studies to obtain deeper or new insights into QWs. In
this paper, we highlight the directivity of one-dimensional discrete quantum
walks via QWRWs. By exploiting the fact that QWRW allows trajectories of
individual walkers to be considered, we first discuss the determination of
future directions of QWRWs, through which the effect of linear spreading and
localization is manifested in another way. Furthermore, the transition
probabilities of QWRWs can also be visualized and show a highly complex shape,
representing QWs in a novel way. Moreover, we discuss the first return time to
the origin between RWs and QWs, which is made possible via the notion of QWRWs.
We observe that the first return time statistics of QWs are quite different
from RWs, caused by both the linear spreading and localization properties of
QWs.
