Random walk on the Poincar disk induced by a group of Mbius transformations.

We consider a discrete-time random motion, Markov chain on the Poincar 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 Mbius transformations. We exploit an isomorphism between the underlying group of Mbius transformations and 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.

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