Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces

TL;DR

This paper models particles moving at finite velocity in hyperbolic spaces, analyzing their branching behavior, mean hyperbolic distance, and how these evolve over time with respect to key parameters.

Contribution

It provides exact expressions for the mean hyperbolic distance of particles' center of mass and explores their behavior in hyperbolic spaces, a novel approach in this context.

Findings

Exact formula for the mean hyperbolic distance of the center of mass.

Analysis of the distance behavior as time increases and parameters vary.

Representation of the distance of a randomly stopped particle on the main geodesic.

Abstract

A branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincar\'e half-plane and Poincar\'e disk) is examined. Each particle can split into two particles only once at Poisson paced times and deviates orthogonally when splitted. At time $t$, after $N(t)$ Poisson events, there are $N(t)+1$ particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as $t$ increases and for different values of the parameters $c$ (hyperbolic velocity of motion) and $\lambda$ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented.