Blocks and Gaps in the Asymmetric Simple Exclusion Process: Asymptotics

TL;DR

This paper derives asymptotic formulas for the probability that a particle in the ASEP is at the start of a block or gap, given its position, in the KPZ regime with step initial conditions.

Contribution

It provides the first asymptotic analysis of block and gap probabilities in ASEP under KPZ scaling, extending previous exact formulas to large-time limits.

Findings

Asymptotic probability for a particle to start an L-block given its position

Asymptotic probability for G unoccupied sites between particles

Results apply in the KPZ regime with step initial conditions

Abstract

In earlier work (arXiv:1707.04927) the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time $t$ a particle is at site $x$ and is the beginning of a block of $L$ consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\rightarrow\infty$) that a particle is the beginning of an $L$-block, given that it is at site $x$ at time $t$. Using duality between occupied and unoccupied sites we obtain the analogous result for a gap of $G$ unoccupied sites between the particle at $x$ and the next one.