Space covering by growing rays

TL;DR

This paper analyzes a one-dimensional space covering process with randomly nucleated rays that grow until collision, deriving exact coverage and distribution results for various nucleation and velocity scenarios.

Contribution

It provides exact analytical results for coverage and distribution properties in a stochastic ray growth process with arbitrary velocity distributions.

Findings

Coverage approaches jammed state as 1/t in continuous nucleation.

Inhomogeneous processes show nonuniversal t^{-p_+} decay.

Simultaneous nucleation depends on velocity distribution details.

Abstract

We study kinetic and jamming properties of a space covering process in one dimension. The stochastic process is defined as follows: Seeds are nucleated randomly in space and produce rays which grow with a constant velocity. The growth stops upon collision with another ray. For arbitrary distributions of the growth velocity, the exact coverage, velocity and size distributions are evaluated for both simultaneous and continuous nucleation. In general, simultaneous nucleation exhibits a stronger dependence on the details of the growth velocity distribution in the asymptotic time regime. The coverage in the continuous case exhibits a universal $1/t$ approach to the jammed state, while an inhomogeneous version of the process leads to nonuniversal $t^{-p_+}$ decay, with $0\le p_+\le 1$ the fraction of right growing rays.