Distributions of a particle's position and their asymptotics in the q-deformed totally asymmetric zero range process with site dependent jumping rates

In this paper we study the probability distribution of the position of a tagged particle in the q-deformed Totally Asymmetric Zero Range Process (q-TAZRP) with site dependent jumping rates. For a finite particle system, it is derived from the transition probability previously obtained by Wang and Waugh. We also provide the probability distribution formula for a tagged particle in the q-TAZRP with the so-called step initial condition in which infinitely many particles occupy one single site and all other sites are unoccupied. For the q-TAZRP with step initial condition, we provide a Fredholm determinant representation for the probability distribution function of the position of a tagged particle, and moreover we obtain the limiting distribution function as the time goes to infinity. Our asymptotic result for q-TAZRP with step initial condition is comparable to the limiting distribution function obtained by Tracy and Widom for the kth leftmost particle in the asymmetric simple exclusion process with step initial condition (Theorem 2 in Tracy and Widom (2009)).