The Transition Probability of the $q$-TAZRP ($q$-Bosons) with Inhomogeneous Jump Rates

TL;DR

This paper derives the transition probability for an inhomogeneous $q$-TAZRP particle system on the integer lattice, generalizing previous homogeneous case results using Bethe ansatz and contour integrals.

Contribution

It provides a new explicit formula for transition probabilities of $q$-TAZRP with site-dependent jump rates, extending prior homogeneous models.

Findings

Derived Bethe ansatz form for transition probabilities

Generalized previous homogeneous $q$-TAZRP results

Expressed probabilities as sums of contour integrals

Abstract

In this paper we consider the $q$-deformed totally asymmetric zero range process ($q$-TAZRP), also known as the $q$-boson (stochastic) particle system, on the ${\mathbb Z}$ lattice, such that the jump rate of a particle depends on the site where it is on the lattice. We derive the transition probability for an $n$ particle process in Bethe ansatz form as a sum of $n!$ $n$-fold contour integrals. Our result generalizes the transition probability formula by Korhonen and Lee for $q$-TAZRP with a homogeneous lattice, and our method follows the same approach as theirs.