## The Annals of Probability

### How to initialize a second class particle?

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3535-3570.

Dates
Revised: August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773658

Digital Object Identifier
doi:10.1214/16-AOP1143

Mathematical Reviews number (MathSciNet)
MR3729609

Zentralblatt MATH identifier
06838101

#### Citation

Balázs, Márton; Nagy, Attila László. How to initialize a second class particle?. Ann. Probab. 45 (2017), no. 6A, 3535--3570. doi:10.1214/16-AOP1143. https://projecteuclid.org/euclid.aop/1511773658

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