Note on a one-dimensional system of annihilating particles

TL;DR

This paper analyzes a one-dimensional particle system with annihilation, proving almost sure annihilation of positive-speed particles on the half-line and survival regimes for zero-speed particles on the full-line under specific speed distributions.

Contribution

It establishes rigorous results on particle annihilation and survival in a stochastic system with symmetric speeds, including new conditions for particle survival.

Findings

Positive-speed particles on the half-line almost surely annihilate.

Zero-speed particles can survive on the full-line under certain speed distributions.

The results depend on the symmetry and discrete nature of the particle speeds.

Abstract

We consider a system of annihilating particles where particles start from the points of a Poisson process on either the full-line or positive half-line and move at constant i.i.d. speeds until collision. When two particles collide, they annihilate. We assume the law of speeds to be symmetric. We prove almost sure annihilation of positive-speed particles started from the positive half-line, and existence of a regime of survival of zero-speed particles on the full-line in the case when speeds can only take 3 values.