  We consider a discrete-time random motion, Markov chain on the Poincar\'{e}
disk. In the basic variant of the model a particle moves along certain circular
arcs within the disk, its location is determined by a composition of random
M\"{o}bius transformations. We exploit an isomorphism between the underlying
group of M\"{o}bius transformations and $\rr$ to study the random motion
through its relation to a one-dimensional random walk. More specifically, we
show that key geometric characteristics of the random motion, such as Busemann
functions and bipolar coordinates evaluated at its location, and hyperbolic
distance from the origin, can be either explicitly computed or approximated in
terms of the random walk. We also consider a variant of the model where the
motion is not confined to a single arc, but rather the particle switches
between arcs of a parabolic pencil of circles at random times.
