From the Riemann surface of TASEP to ASEP

TL;DR

This paper develops a method to connect the Bethe roots of TASEP and ASEP using Laurent series, enabling a new integral representation of ASEP height probabilities on a Riemann surface.

Contribution

It introduces a formal Laurent series mapping that decouples ASEP Bethe equations at all orders in q, linking TASEP and ASEP solutions.

Findings

Decoupling of ASEP Bethe equations via Laurent series.

Representation of ASEP height probabilities as a contour integral.

Connection between TASEP and ASEP Bethe roots through a Riemann surface.

Abstract

We consider the asymmetric simple exclusion process (ASEP) with forward hopping rate 1, backward hopping rate q and periodic boundary conditions. We show that the Bethe equations of ASEP can be decoupled, at all order in perturbation in the variable q, by introducing a formal Laurent series mapping the Bethe roots of the totally asymmetric case q=0 (TASEP) to the Bethe roots of ASEP. The probability of the height for ASEP is then written as a single contour integral on the Riemann surface on which symmetric functions of TASEP Bethe roots live.