Cutoff profile of ASEP on a segment

TL;DR

This paper characterizes the precise cutoff profile and window for ASEP on a segment, revealing a Tracy-Widom distribution connection and providing a new proof of the cutoff phenomenon.

Contribution

It establishes the cutoff window and profile for ASEP on a segment, linking it to Tracy-Widom fluctuations and introducing algebraic identities for analysis.

Findings

Cutoff window of order N^{1/3} for ASEP on a segment.

Cutoff profile given by 1 - Tracy-Widom GUE distribution.

Provides a new proof of the ASEP cutoff phenomenon.

Abstract

This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length $N$. Our main result is that for particle densities in $(0,1),$ the total-variation cutoff window of ASEP is $N^{1/3}$ and the cutoff profile is $1-F_{\mathrm{GUE}},$ where $F_{\mathrm{GUE}}$ is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labb\'{e} and Lacoin. Our proof combines coupling arguments, the result of Tracy-Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra.