Limit law of a second class particle in TASEP with non-random initial condition

TL;DR

This paper studies the fluctuations of a second class particle in TASEP with non-random initial conditions, revealing a $t^{1/3}$ scale fluctuation and its limiting distribution, especially across shock regions.

Contribution

It establishes the limit law for the second class particle's position in TASEP with non-random initial conditions, including the fluctuation scale and distribution.

Findings

Second class particle fluctuates on a $t^{1/3}$ scale.

Limiting distribution of the particle's position is determined.

Distribution of the number of steps taken by the particle is obtained.

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density $\rho$ on $\mathbb{Z}_-$ and $\lambda$ on $\mathbb{Z}_+$, and a second class particle initially at the origin. For $\rho<\lambda$, there is a shock and the second class particle moves with speed $1-\lambda-\rho$. For large time $t$, we show that the position of the second class particle fluctuates on a $t^{1/3}$ scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time $t$.