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

Eunghyun Lee; Dong Wang

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arXiv · 1703.08839

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

Eunghyun Lee·Dong Wang

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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)).

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BibTeX RIS

Eunghyun Lee, Dong Wang. 2017-03-26. Distributions of a particle's position and their asymptotics in the $q$-deformed totally asymmetric zero range process with site dependent jumping rates. https://arxiv.org/abs/1703.08839

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