### 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 |
---|---|

Article number | 013301 |

Journal | Journal of Mathematical Physics |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*55*(1), [013301]. https://doi.org/10.1063/1.4851758

**The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process.** / Korhonen, Marko; Lee, Eunghyun.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 55, no. 1, 013301. https://doi.org/10.1063/1.4851758

}

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

PY - 2014

Y1 - 2014

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.

UR - http://www.scopus.com/inward/record.url?scp=84902338962&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902338962&partnerID=8YFLogxK

U2 - 10.1063/1.4851758

DO - 10.1063/1.4851758

M3 - Article

VL - 55

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

M1 - 013301

ER -