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 non-concave, non-convex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of P. A. Ferrari and C. Kipnis' arguments ("Second class particles in the rarefaction fan", Ann. Inst. H. Poincaré, 31, 1995) 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. Keywords. second class particle, limit distribution, rarefaction fan, shock, hydrodynamic limit, collision probability. Acknowledgement.The authors thank valuable discussions with Pablo A. Ferrari on the problem, with Ellen Saada on the hydrodynamic limit of asymmetric processes and with Cédric Bernardin on symmetric processes. We also thank the anonymous referee for his/her comments. M. Balázs acknowledges support from the Hungarian National Research, Development and Innovation Office, NKFIH grants K100473 and K109684.

Limiting speed of a second class particle in ASEP

“…Tracy and Widom [2009a] found the exact distribution of the second class particle in the ASEP. Balázs and Nagy [2017] generalized the result of Ferrari and Kipnis [1995] for a wide range of partially and totally asymmetric interacting particles systems.…”

Coloured stochastic vertex models and their spectral theory

“…Its extensions to many colours for the ASEP with step initial condition have been investigated by Amir-Angel-Valko [AAV11] via a matching result that is not far from (the ASEP degeneration of) ours. For a single second class particle, a recent work by Balázs-Nagy [BN17] proves results for more general (continuous time) processes, again by a suitable matching.…”