Distributions of a particle's position and their asymptotics in the -deformed totally asymmetric zero range process with site dependent jumping rates
arXiv:1703.08839
Abstract
In this paper we study the probability distribution of the position of a tagged particle in the -deformed Totally Asymmetric Zero Range Process (-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 -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 -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 -TAZRP with step initial condition is comparable to the limiting distribution function obtained by Tracy and Widom for the -th leftmost particle in the asymmetric simple exclusion process with step initial condition (Theorem 2 in Commun. Math. Phys. 290, 129--154 (2009)).
34 pages, 2 figures