# Cutoff and discrete Product Structure in ASEP

## TL;DR

This paper studies the ASEP with a discontinuous initial density profile, revealing a cutoff phenomenon at the discontinuity and a discrete product limit law for particle fluctuations, especially in the totally ASEP case.

## Contribution

It introduces a cutoff law for particle positions at the discontinuity and identifies a discrete product limit law within the discontinuity region, extending understanding of ASEP fluctuations.

## Key findings

- Limit law of particle position exhibits cutoff under t^{1/2} scaling.
- Discrete product limit law bounds fluctuations inside the discontinuity.
- In totally ASEP, the limit law matches the fluctuations exactly.

## Abstract

We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ with an initial data such that in the large time particle density $\rho(\cdot)$ a discontinuity at the origin is created, where the value of $\rho$ jumps from zero to one, but $\rho(-\varepsilon),1-\rho(\varepsilon) >0 $ for any $\varepsilon>0$. We consider the position of a particle $x_{M}$ macroscopically located at the discontinuity, and show that its limit law has a cutoff under $t^{1/2}$ scaling. Inside the discontinuity region, we show that a discrete product limit law arises, which bounds from above the limiting fluctuations of $x_{M}$ in the general ASEP, and equals them in the totally ASEP. Note: This preprint has been superseded by arXiv:1906.07711 and is no longer updated.