Random walk on the Poincaré disk induced by a group of Möbius transformations.

IF 0.4 4区 数学 Q4 STATISTICS & PROBABILITY
Markov Processes and Related Fields Pub Date : 2019-01-01
Charles McCarthy, Gavin Nop, Reza Rastegar, Alexander Roitershtein
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引用次数: 0

Abstract

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 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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由一组Möbius变换引起的庞卡罗圆盘上的随机漫步。
我们考虑一个离散时间随机运动,庞卡卡尔圆盘上的马尔可夫链。在该模型的基本变体中,粒子沿着圆盘内的某些圆弧运动,其位置由随机Möbius变换的组合决定。我们利用底层的Möbius变换群与l之间的同构关系,通过它与一维随机游走的关系来研究随机运动。更具体地说,我们表明随机运动的关键几何特征,如Busemann函数和在其位置评估的双极坐标,以及到原点的双曲线距离,可以通过随机行走显式计算或近似。我们还考虑了模型的一种变体,其中运动不局限于单个弧,而是粒子在随机时间在抛物线形圆铅笔的弧之间切换。
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来源期刊
Markov Processes and Related Fields
Markov Processes and Related Fields STATISTICS & PROBABILITY-
CiteScore
0.70
自引率
0.00%
发文量
0
期刊介绍: Markov Processes And Related Fields The Journal focuses on mathematical modelling of today''s enormous wealth of problems from modern technology, like artificial intelligence, large scale networks, data bases, parallel simulation, computer architectures, etc. Research papers, reviews, tutorial papers and additionally short explanations of new applied fields and new mathematical problems in the above fields are welcome.
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