Title: Distributions of a particle's position and their asymptotics in the $q$-deformed totally asymmetric zero range process with site dependent jumping rates,
Abstract: 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 $k$-th leftmost
particle in the asymmetric simple exclusion process with step initial condition
(Theorem 2 in Commun. Math. Phys. 290, 129--154 (2009)).