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

Abstract

We treat the $N$-particle ZRP 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 that appeared in [J. Phys. A, \textbf{31} 6057--6071 (1998)] by Sasamoto and Wadati or the $q$-TAZRP in \textit{MacDonald processes} by Borodin and Corwin. 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 Tracy and Widom's identity for the ASEP in [Commun. Math. Phys., \textbf{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.

Added remarks and references, corrected minor errors

References in corpus (10)

• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• From duality to determinants for q-TASEP and ASEP
• Current Distribution and random matrix ensembles for an integrable asymmetric fragmentation process
• On integrability of zero-range chipping models with factorized steady state
• Bethe ansatz solution of zero-range process with nonuniform stationary state
• The Asymmetric Avalanche Process
• Bethe ansatz and current distribution for the TASEP with particle-dependent hopping rates
• Determinant representation for some transition probabilities in the TASEP with second class particles
• The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension

Cited by in corpus (6)

• Tracy-Widom asymptotics for q-TASEP
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Probability distributions of multi-species q-TAZRP and ASEP as double cosets of parabolic subgroups
• Current statistics in the q-boson zero range process
• Some conditional probabilities in the TASEP with second class particles
• Q-zero range has random walking shocks