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The transition probability and the probability for the left-most particle's position of the

*q*-totally asymmetric zero range process

### 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. A31, 6057–6071 (1998)] or the q-totally 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 new identity corresponding to identity for the asymmetric simple exclusion process by Tracy and Widom [“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.

© 2014 AIP Publishing LLC

Received Tue Sep 17 00:00:00 UTC 2013
Accepted Fri Dec 06 00:00:00 UTC 2013
Published online Wed Jan 15 00:00:00 UTC 2014

Acknowledgments:
The authors would like to thank Alexei Borodin and Ivan Corwin for valuable comments on the earlier version of the manuscript. We also would like to express our gratitude to Antti Kupiainen for helpful comments and unstinting support. E. Lee was partially supported by European Research Council Advanced Grant, the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec – nature et technologies (FQRNT), and the CRM Laboratoire de Physique Mathématique. M. Korhonen was supported by Academy of Finland.

Article outline:

I. INTRODUCTION
A. Bethe ansatz applicability
B. Transition probability and the left-most particle's position at time *t*
C. Outline
II. TRANSITION PROBABILITY OF THE *q*-TAZRP
A. Preliminary
1. 2-particle ZRP
2. 3-particle ZRP
3. *N*-particle ZRP
B. Transition probability
C. On the mapping between the *q*-TAZRP and the *q*-TASEP
III. THE PROBABILITY FOR THE LEFT-MOST PARTICLE'S POSITION

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2014-01-15

2016-12-10

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