TY - JOUR

T1 - The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process

AU - Korhonen, Marko

AU - Lee, Eunghyun

N1 - Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2014/1

Y1 - 2014/1

N2 - We treat the N-particle zero range process whose jumping rates satisfy a certain condition. This condition is required to use the Bethe ansatz and the resulting model is the q-boson model by Sasamoto and Wadati ["Exact results for one-dimensional totally asymmetric diffusion models," J. Phys. A 31, 6057-6071 (1998)] or the qtotally asymmetric zero range process (TAZRP) by Borodin and Corwin ["Macdonald processes," Probab. Theory Relat. Fields (to be published)]. We find the explicit formula of the transition probability of the q-TAZRP via the Bethe ansatz. By using the transition probability we find the probability distribution of the left-most particle's position at time t. To find the probability for the left-most particle's position we find a newidentity corresponding to identity for the asymmetric simple exclusion process by Tracy andWidom ["Integral formulas for the asymmetric simple exclusion process," Commun. Math. Phys. 279, 815-844 (2008)]. For the initial state that all particles occupy a single site, the probability distribution of the left-most particle's position at time t is represented by the contour integral of a determinant.

AB - We treat the N-particle zero range process whose jumping rates satisfy a certain condition. This condition is required to use the Bethe ansatz and the resulting model is the q-boson model by Sasamoto and Wadati ["Exact results for one-dimensional totally asymmetric diffusion models," J. Phys. A 31, 6057-6071 (1998)] or the qtotally asymmetric zero range process (TAZRP) by Borodin and Corwin ["Macdonald processes," Probab. Theory Relat. Fields (to be published)]. We find the explicit formula of the transition probability of the q-TAZRP via the Bethe ansatz. By using the transition probability we find the probability distribution of the left-most particle's position at time t. To find the probability for the left-most particle's position we find a newidentity corresponding to identity for the asymmetric simple exclusion process by Tracy andWidom ["Integral formulas for the asymmetric simple exclusion process," Commun. Math. Phys. 279, 815-844 (2008)]. For the initial state that all particles occupy a single site, the probability distribution of the left-most particle's position at time t is represented by the contour integral of a determinant.

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U2 - 10.1063/1.4851758

DO - 10.1063/1.4851758

M3 - Article

AN - SCOPUS:84902338962

VL - 55

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

M1 - 013301

ER -