The upper threshold in ballistic annihilation

TL;DR

This paper investigates the critical probability threshold in a three-speed ballistic annihilation model, improving bounds and employing renewal properties and computational methods to approach a long-standing conjecture.

Contribution

It introduces a renewal approach that relates particle survival to Galton-Watson processes and improves the upper bound on the critical probability for the integer-start variant.

Findings

Improved upper bound on critical probability to 0.2870

Renewal property links particle survival to Galton-Watson processes

Potential to address the conjecture that p_c > 0

Abstract

Three-speed ballistic annihilation starts with infinitely many particles on the real line. Each is independently assigned either speed-$0$ with probability $p$, or speed-$\pm 1$ symmetrically with the remaining probability. All particles simultaneously begin moving at their assigned speeds and mutually annihilate upon colliding. Physicists conjecture when $p \leq p_c = 1/4$ all particles are eventually annihilated. Dygert et. al. prove $p_c \leq .3313$, while Sidoravicius and Tournier describe an approach to prove $p_c \leq .3281$. For the variant in which particles start at the integers, we improve the bound to $.2870$. A renewal property lets us equate survival of a particle to the survival of a Galton-Watson process whose offspring distribution a computer can rigorously approximate. This approach may help answer the nearly thirty-year old conjecture that $p_c >0$.