Random walk on the Poincar\'{e} disk induced by a group of M\"{o}bius transformations

TL;DR

This paper studies a Markov chain on the Poincaré disk driven by random Möbius transformations, linking its geometric properties to a one-dimensional random walk and analyzing a variant with switching arcs.

Contribution

It introduces a novel analysis of a random walk on the Poincaré disk via Möbius transformations and relates geometric characteristics to a simpler one-dimensional random walk.

Findings

Explicit computation of hyperbolic distance and geometric functions

Approximation methods for geometric characteristics

Analysis of a variant with arc-switching dynamics

Abstract

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.