Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 037, 16 pages      arXiv:1512.01612      https://doi.org/10.3842/SIGMA.2016.037
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

The Transition Probability of the $q$-TAZRP ($q$-Bosons) with Inhomogeneous Jump Rates

Dong Wang a and David Waugh b
a) Department of Mathematics, National University of Singapore, 119076, Singapore
b) BNP Paribas, 787 7th Avenue, New York, NY, 10019, USA

Received January 03, 2016, in final form April 08, 2016; Published online April 14, 2016

Abstract
In this paper we consider the $q$-deformed totally asymmetric zero range process ($q$-TAZRP), also known as the $q$-boson (stochastic) particle system, on the ${\mathbb Z}$ lattice, such that the jump rate of a particle depends on the site where it is on the lattice. We derive the transition probability for an $n$ particle process in Bethe ansatz form as a sum of $n!$ $n$-fold contour integrals. Our result generalizes the transition probability formula by Korhonen and Lee for $q$-TAZRP with a homogeneous lattice, and our method follows the same approach as theirs.

Key words: zero range process; transition probability; interacting particle systems; Bethe ansatz.

pdf (434 kb)   tex (32 kb)

References

  1. Balázs M., Komjáthy J., Order of current variance and diffusivity in the rate one totally asymmetric zero range process, J. Stat. Phys. 133 (2008), 59-78, arXiv:0804.1397.
  2. Borodin A., Corwin I., Macdonald processes, Probab. Theory Related Fields 158 (2014), 225-400, arXiv:1111.4408.
  3. Borodin A., Corwin I., Petrov L., Sasamoto T., Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz, Comm. Math. Phys. 339 (2015), 1167-1245, arXiv:1407.8534.
  4. Borodin A., Corwin I., Petrov L., Sasamoto T., Spectral theory for the $q$-Boson particle system, Compos. Math. 151 (2015), 1-67, arXiv:1308.3475.
  5. Borodin A., Corwin I., Sasamoto T., From duality to determinants for $q$-TASEP and ASEP, Ann. Probab. 42 (2014), 2314-2382, arXiv:1207.5035.
  6. Borodin A., Petrov L., Higher spin six vertex model and symmetric rational functions, arXiv:1601.05770.
  7. Corwin I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76 pages, arXiv:1106.1596.
  8. Gwa L.-H., Spohn H., Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian, Phys. Rev. Lett. 68 (1992), 725-728.
  9. Korhonen M., Lee E., The transition probability and the probability for the left-most particle's position of the $q$-totally asymmetric zero range process, J. Math. Phys. 55 (2014), 013301, 15 pages, arXiv:1308.4769.
  10. Lee E., Transition probabilities of the Bethe ansatz solvable interacting particle systems, J. Stat. Phys. 142 (2011), 643-656, arXiv:1011.2841.
  11. Liggett T.M., Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005.
  12. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  13. Povolotsky A.M., On the integrability of zero-range chipping models with factorized steady states, J. Phys. A: Math. Theor. 46 (2013), 465205, 25 pages, arXiv:1308.3250.
  14. Quastel J., Introduction to KPZ, in Current Developments in Mathematics, 2011, Int. Press, Somerville, MA, 2012, 125-194.
  15. Sasamoto T., Wadati M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A: Math. Gen. 31 (1998), 6057-6071.
  16. Schütz G.M., Exact solution of the master equation for the asymmetric exclusion process, J. Stat. Phys. 88 (1997), 427-445, cond-mat/9701019.
  17. Spitzer F., Interaction of Markov processes, Adv. Math. 5 (1970), 246-290.
  18. Tracy C.A., Widom H., Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815-844, arXiv:0704.2633.
  19. Tracy C.A., Widom H., Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129-154, arXiv:0807.1713.

Previous article  Next article   Contents of Volume 12 (2016)