The ballistic annihilation threshold is positive

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.