The ballistic annihilation threshold is positive

TL;DR

This paper proves that in ballistic annihilation, the critical stationary particle proportion for phase transition is at least approximately 0.217, establishing a nontrivial lower bound for the survival threshold.

Contribution

It provides the first rigorous lower bound on the phase transition point in ballistic annihilation, improving understanding of the survival probability regime.

Findings

Proves that the critical threshold p_c is at least 0.21699.

Establishes a comparable bound for a discretised version of the process.

Advances the theoretical understanding of phase transition in ballistic annihilation.

Abstract

In the ballistic annihilation process, particles on the real line have independent speeds symmetrically distributed in $\{-1,0,+1\}$ and are annihilated by collisions. It is widely believed that there is a phase transition at $p=p_{\mathrm c}=0.25$ between regimes where every particle is eventually annihilated and where some particles survive forever, where $p$ is the proportion of stationary particles. It is easy to see that some particles survive if $p>0.5$, and rigorous proofs giving better upper bounds on $p_{\mathrm c}$ have recently appeared. However, no nontrivial lower bound on $p_{\mathrm c}$ was previously known. We prove that $p_{\mathrm c}\geq 0.21699$, and give a comparable bound for a discretised version.