TY - JOUR

T1 - Distributions of a particle's position and their asymptotics in the q-deformed totally asymmetric zero range process with site dependent jumping rates

AU - Lee, Eunghyun

AU - Wang, Dong

N1 - Funding Information:
The authors thank the helpful discussion with Guillaume Barraquand. The first author was supported partially by the social policy grant and the faculty-development competitive grant 090118FD5341 of Nazarbayev University . The second author was supported partially by the Singapore AcRF Tier 1 grant R-146-000-217-112 .

PY - 2019/5

Y1 - 2019/5

N2 - 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)).

AB - 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)).

KW - Coordinate Bethe ansatz

KW - Integrable probability

KW - Limiting distribution

KW - q-deformed totally asymmetric zero range process

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U2 - 10.1016/j.spa.2018.06.005

DO - 10.1016/j.spa.2018.06.005

M3 - Article

AN - SCOPUS:85049469878

VL - 129

SP - 1795

EP - 1828

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 5

ER -