Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass

TL;DR

This paper investigates the behavior of a particle moving randomly on the Poincaré half-plane, analyzing its mean hyperbolic distance over time, including cases with returns to the start, and compares it with motion on spherical circles.

Contribution

It introduces a model of random geodesic motion on the Poincaré half-plane and derives explicit formulas for the mean hyperbolic distance, including return scenarios and spherical analogs.

Findings

Derived formulas for mean hyperbolic distance in various motion scenarios

Analyzed the effect of returns to the starting point on distance evolution

Compared hyperbolic motion with spherical motion on orthogonal circles

Abstract

A random motion on the Poincar\'e half-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyperbolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated.