Probability distributions of multi-species q-TAZRP and ASEP as double cosets of parabolic subgroups

TL;DR

This paper derives explicit contour integral formulas for the probability distributions of multi-species q-TAZRP and ASEP starting from q-exchangeable initial conditions, extending single-species results through double coset decompositions.

Contribution

It introduces a novel approach using double coset decomposition for multi-species q-TAZRP and provides explicit formulas for multi-species ASEP, generalizing known single-species formulas.

Findings

Explicit contour integral formulas for multi-species q-TAZRP and ASEP

Use of double coset decomposition for multi-species q-TAZRP

Direct proof for multi-species ASEP formulas

Abstract

We write explicit contour integral formulas for probability distributions of the multi-species q-TAZRP and the multi-species ASEP starting with q-exchangeable initial conditions. The formulas are equal to the corresponding explicit contour integral formulas for the single-species q-TAZRP ([Korhonen-Lee 2014, Wang-Waugh 2016]) and ASEP [Tracy-Widom 2007], with a factor in front of the integral. For the multi-species q-TAZRP, we use a decomposition theorem for elements of double cosets of parabolic subgroups in a Coxeter group. The set of distinguished double coset representatives with minimal length is viewed as a particle configuration. For the multi-species ASEP we use a more direct proof.