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The phase structure of asymmetric ballistic annihilation. (English) Zbl 1498.60388

Summary: 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 J. Haslegrave et al. [Sel. Math., New Ser. 27, No. 5, Paper No. 84, 38 p. (2021; Zbl 1470.60279)]. 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 1470.60279
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References:

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