由一组Möbius变换引起的Poincaré圆盘上的随机游动
Random walk on the Poincaré disk induced by a group of Möbius transformations
摘要：
考虑Poincaré圆盘上的离散时间随机运动马尔可夫链。在该模型的基本变体中，一个粒子沿着圆盘内的某些圆弧运动，它的位置由随机莫比乌斯变换的组合决定。我们利用一个同构之间的基础组的莫比乌斯变换和$\rr$研究随机运动通过其关系的一维随机游动。更具体地说，我们表明，关键的几何特征的随机运动，如Busemann函数和双极坐标在其位置评估，从原点和双曲距离，可以明确计算或近似的随机游动。我们还考虑了一个变体的模型，其中的运动不限于一个单一的弧，而是在随机时间的抛物线圆束的弧之间的粒子开关。
英文摘要：
We consider a discrete-time random motion, Markov chain on the Poincaré 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 Möbius transformations. We exploit an isomorphism between the underlying group of Möbius 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.
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