Motion among random obstacles on a hyperbolic space

TL;DR

This paper studies the motion of a particle in a hyperbolic space with randomly placed obstacles, showing that as obstacle density increases and size decreases, the particle's behavior converges to a hyperbolic random flight.

Contribution

It introduces a hyperbolic analogue of the Lorentz process and proves convergence to a Markovian hyperbolic random flight under specific scaling limits.

Findings

Convergence of the particle process to a hyperbolic Markovian flight.

Extension of Lorentz process analysis to hyperbolic geometry.

Establishment of a scaling limit in hyperbolic space.

Abstract

We consider the motion of a particle along the geodesic lines of the Poincar\`e half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied by Gallavotti in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.