Atypical behaviors of a tagged particle in asymmetric simple exclusion

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 $N\uparrow \mathrm{\infty }$. In this context, we study the typical' behavior of the tagged particle and bulk’ density evolution subject to atypical' events $\{X_N\geq AN\}$ or $\{X_N\leq AN\}$ for $A\neq γ(1-ρ)$. We detail different structures, depending on whether $A<0$, $0\leq A< γ(1-ρ)$, $γ(1-ρ)<A< γ$, or $A\geq γ$, 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 $\frac{1}{N}{X}_N$.