Atypical behaviors of a tagged particle in asymmetric simple exclusion

Abstract

Consider the asymmetric nearest-neighbor exclusion process (ASEP) on $\mathbb{Z}$ with single particle drift $\gamma >0$, starting from a Bernoulli product invariant measure ${\nu }_{\rho }$ with density $\rho$. It is known that the position $X_N$ of a tagged particle, say initially at the origin, at time $N$ satisfies an a.s. law of large numbers $\frac{1}{N}{X}_N\to \gamma (1-\rho )$ as $N\uparrow \infty$. In this context, we study the `typical' behavior of the tagged particle and `bulk' density evolution subject to `atypical' events $\{X_N\ge AN\}$ or $\{X_N\le AN\}$ for $A\ne \gamma (1-\rho )$. We detail different structures, depending on whether $A<0$, $0\le A<\gamma (1-\rho )$, $\gamma (1-\rho )<A<\gamma$, or $A\ge \gamma$, under which these atypical events are achieved, and compute associated large deviation costs. Among our results is an `upper tail' large deviation principle in scale $N$ for $\frac{1}{N}{X}_N$.

39 pages