Collision probabilities in the rarefaction fan of asymmetric exclusion processes

TL;DR

This paper analyzes the collision probabilities of second-class particles in the asymmetric simple exclusion process (ASEP), providing explicit formulas and exploring implications for hydrodynamic limits and multi-type particle systems.

Contribution

It derives an explicit probability for the collision of two second-class particles in ASEP and connects this to hydrodynamic limits and multi-type particle representations.

Findings

Collision probability for two second-class particles is (1+p)/3p.

In the totally asymmetric case, a multi-type particle system representation is provided.

Results inform the probability of cluster coexistence in a two-type corner growth model.

Abstract

We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate $p\in(1/2,1]$ and to the left at rate $1-p$, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and 1, and holes to the right of site 1. We show that the probability that the two second-class particles eventually collide is $(1+p)/3p$, where a_collision_ occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes, started from appropriate initial states and coupled using the so-called "basic coupling", eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case ($p=1$) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model.