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

Dong Wang a and David Waugh b
a) Department of Mathematics, National University of Singapore, 119076, Singapore
b) BNP Paribas, 787 7th Avenue, New York, NY, 10019, USA

Received January 03, 2016, in final form April 08, 2016; Published online April 14, 2016

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.

Key words: zero range process; transition probability; interacting particle systems; Bethe ansatz.

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