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November 2017 How to initialize a second class particle?
Márton Balázs, Attila László Nagy
Ann. Probab. 45(6A): 3535-3570 (November 2017). DOI: 10.1214/16-AOP1143


We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile, which in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with nonconcave, nonconvex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of Ferrari and Kipnis’ arguments (Ann. Inst. H. Poincaré 31 (1995) 143–154) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle–antiparticle pair.


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Márton Balázs. Attila László Nagy. "How to initialize a second class particle?." Ann. Probab. 45 (6A) 3535 - 3570, November 2017.


Received: 1 January 2016; Revised: 1 August 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838101
MathSciNet: MR3729609
Digital Object Identifier: 10.1214/16-AOP1143

Primary: 60K35
Secondary: 82C22

Keywords: collision probability , Hydrodynamic limit , limit distribution , Rarefaction fan , Second class particle , Shock

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 6A • November 2017
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