Two ways to solve ASEP

TL;DR

This paper compares two methods for deriving exact formulas for particle current distribution in ASEP with step initial data, highlighting their theoretical foundations and differences.

Contribution

It details two distinct approaches—Bethe ansatz and replica trick—for computing exact formulas in ASEP, clarifying their theoretical underpinnings.

Findings

Both methods yield exact formulas for ASEP current distribution.

The Bethe ansatz approach is based on coordinate Bethe ansatz techniques.

The replica trick approach uses a rigorous version of the replica method.

Abstract

The purpose of this article is to describe the two approaches to compute exact formulas (which are amenable to asymptotic analysis) for the probability distribution of the current of particles past a given site in the asymmetric simple exclusion process (ASEP) with step initial data. The first approach is via a variant of the coordinate Bethe ansatz and was developed in work of Tracy and Widom in 2008-2009, while the second approach is via a rigorous version of the replica trick and was developed in work of Borodin, Sasamoto and the author in 2012.