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In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusCombinatorial Universality in Three-Speed Ballistic Annihilation

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Combinatorial Universality in... [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusCombinatorial Universality in Three-Speed Ballistic Annihilation

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.

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References (16)

Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-60753-1
Pages
487 –517
DOI
10.1007/978-3-030-60754-8_23
Publisher site
See Chapter on Publisher Site

Abstract

[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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