Busemann functions and the speed of a second class particle in the rarefaction fan

TL;DR

This paper leverages Busemann functions from last-passage percolation to determine the asymptotic speed distribution of a second class particle in systems with a rarefaction fan, such as exclusion and Hammersley processes.

Contribution

It introduces a novel method connecting Busemann functions to the asymptotic behavior of second class particles in complex particle systems.

Findings

Derived the distribution of the second class particle's speed in the rarefaction fan

Extended the application of Busemann functions to particle system dynamics

Connected last-passage percolation results to particle system behavior

Abstract

In this paper we will show how the results found in Cator and Pimentel 2009, about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process, or the Hammersley interacting particle process. The method will be to use the well known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well known connection between second class particles and competition interfaces.