The phase structure of asymmetric ballistic annihilation

Abstract

Ballistic annihilation is an interacting system in which particles placed throughout the real line move at preassigned velocities and annihilate upon colliding. The longstanding conjecture that in the symmetric three-velocity setting there exists a phase transition for the survival of middle-velocity particles was recently resolved by Haslegrave, Sidoravicius, and Tournier. We develop a framework based on a mass transport principle to analyze three-velocity ballistic annihilation with asymmetric velocities assigned according to an asymmetric probability measure. We show the existence of a phase transition in all cases by deriving universal bounds. In particular, all middle-speed particles perish almost surely if their initial density is less than $1/5$, regardless of the velocities, relative densities, and spacing of initial particles. We additionally prove the continuity of several fundamental statistics as the probability measure is varied.