### Abstract

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.

Original language | English |
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Article number | 013301 |

Journal | Journal of Mathematical Physics |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2014 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics