October 2022 The phase structure of asymmetric ballistic annihilation
Matthew Junge, Hanbaek Lyu
Author Affiliations +
Ann. Appl. Probab. 32(5): 3797-3816 (October 2022). DOI: 10.1214/21-AAP1773


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.

Funding Statement

Junge was partially supported by NSF Grant DMS #1855516 and Lyu was partially supported by NSF Grant DMS #2010035.


The authors appreciate valuable comments from Rick Durrett and Tom Liggett.


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Matthew Junge. Hanbaek Lyu. "The phase structure of asymmetric ballistic annihilation." Ann. Appl. Probab. 32 (5) 3797 - 3816, October 2022. https://doi.org/10.1214/21-AAP1773


Received: 1 March 2021; Revised: 1 November 2021; Published: October 2022
First available in Project Euclid: 18 October 2022

MathSciNet: MR4497858
zbMATH: 1498.60388
Digital Object Identifier: 10.1214/21-AAP1773

Primary: 60K35

Keywords: Interacting particle system , phase transition

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 5 • October 2022
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