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(2021)
Probability and random processesDigital Signal Processing
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(2014)
Ponder This
J. Haslegrave, V. Sidoravicius, L. Tournier (2018)
The three-speed ballistic annihilation threshold is 1/4arXiv: Probability
B. Dygert, C. Kinzel, Jennifer Zhu, M. Junge, Annie Raymond, Erik Slivken (2016)
The bullet problem with discrete speedsElectronic Communications in Probability
[We consider a one-dimensional system of particles, moving at constant velocities chosen independently according to a symmetric distribution on {−1, 0, +1}, and annihilating upon collision—with, in case of triple collision, a uniformly random choice of survivor among the two moving particles. When the system contains infinitely many particles, whose starting locations are given by a renewal process, a phase transition was proved to happen (see Haslegrave et al., Three-speed ballistic annihilation: phase transition and universality, 2018) as the density of static particles crosses the value 1∕4. Remarkably, this critical value, along with certain other statistics, was observed not to depend on the distribution of interdistances. In the present paper, we investigate further this universality by proving a stronger statement about a finite system of particles with fixed, but randomly shuffled, interdistances. We give two proofs, one by an induction allowing explicit computations, and one by a more direct comparison. This result entails a new nontrivial independence property that in particular gives access to the density of surviving static particles at time t in the infinite model. Finally, in the asymmetric case, further similar independence properties are proved to keep holding, including a striking property of gamma distributed interdistances that contrasts with the general behavior.]
Published: Nov 4, 2020
Keywords: Ballistic annihilation; Interacting particle system; Random permutation; Gamma distribution; 60K35
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