Limiting speed of a second class particle in ASEP

TL;DR

This paper investigates the asymptotic speed distribution of a second class particle in a two-species ASEP with specific initial conditions, revealing it converges to a distribution related to the minimum of uniform samples.

Contribution

It characterizes the limiting distribution of the second class particle's speed in ASEP with particular initial data, connecting it to the minimum of uniform random variables.

Findings

Speed converges to a distribution supported on a symmetric interval.

Limiting distribution equals the minimum of L+1 independent uniform samples.

Provides a probabilistic description of second class particle dynamics.

Abstract

We study the asymptotic speed of a second class particle in the two-species asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ with each particle belonging either to the first class or the second class. For any fixed non-negative integer $L$, we consider the two-species ASEP started from the initial data with all the sites of $\mathbb{Z}_{<-L}$ occupied by first class particles, all the sites of $\mathbb{Z}_{[-L,0]}$ occupied by second class particles, and the rest of the sites of $\mathbb{Z}$ unoccupied. With these initial conditions, we show that the speed of the leftmost second class particle converges weakly to a distribution supported on a symmetric compact interval $\Gamma\subset \mathbb{R}$. Furthermore, the limiting distribution is shown to have the same law as the minimum of $L+1$ independent random samples drawn uniformly from the interval $\Gamma$.