# Collision probabilities in the rarefaction fan of asymmetric exclusion processes - Mathematics > Probability

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Abstract: We consider the one-dimensional asymmetric simple exclusion process ASEP inwhich particles jump to the right at rate $p\in1-2,1$ and to the left at rate$1-p$, interacting by exclusion. In the initial state there is a finite regionsuch that to the left of this region all sites are occupied and to the right ofit all sites are empty. Under this initial state, the hydrodynamical limit ofthe process converges to the rarefaction fan of the associated Burgersequation. In particular suppose that the initial state has first-classparticles 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 twosecond-class particles eventually collide is $1+p-3p$, where a collision occurs when one of the particles attempts to jump over the other. This alsocorresponds to the probability that two ASEP processes, started fromappropriate initial states and coupled using the so-called -basic coupling-,eventually reach the same state. We give various other results about thebehaviour 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 particlesystem, and also use the collision result to derive the probability ofcoexistence of both clusters in a two-type version of the corner growth model.

Author: ** Pablo A. Ferrari, Patricia Goncalves, James B. Martin**

Source: https://arxiv.org/