Advertisement

Probability Distributions of Multi-species q-TAZRP and ASEP as Double Cosets of Parabolic Subgroups

  • Jeffrey KuanEmail author
Article
  • 6 Downloads

Abstract

We write explicit contour integral formulas for probability distributions of the multi-species q-TAZRP and the multi-species ASEP starting with q-exchangeable initial conditions. The formulas are equal to the corresponding explicit contour integral formulas for the single-species q-TAZRP (Korhonen and Lee in J Math Phys 55:013301, 2014. arXiv:1308.4769v2, Wang and Waugh in SIGMA 12:037, 2016. arXiv:1512.01612v5) and ASEP (Tracy and Widom in Integral formulas for the asymmetric simple exclusion process, 2007. arXiv:0704.2633), with a factor in front of the integral. For the multi-species q-TAZRP, we use a decomposition theorem for elements of double cosets of parabolic subgroups in a Coxeter group. The set of distinguished double coset representatives with minimal length is viewed as a particle configuration. For the multi-species ASEP, we use a more direct proof.

Mathematics Subject Classification

Primary 05E15 Secondary 60C05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Andjel, E.D.: Invariant measures for the zero range process. Ann. Probab. 10, 525–547 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  3. 3.
    Arita, C.: Remarks on the multi-species exclusion process with reflective boundaries. J. Phys. A Math. Theor. (2012).  https://doi.org/10.1088/1751-8113/45/15/155001
  4. 4.
    Belitsky, V., Schütz, G.M.: Self-duality and shock dynamics in the \(n\)-component priority ASEP. arXiv:1606.04587v1
  5. 5.
    Billey, S., Konvalinka, M., Petersen, T., Slofstra, W., Tenner, B.: Parabolic double cosets in Coxeter groups. Electron. J. Comb. arXiv:1612.00736v2
  6. 6.
    Carter, R.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley, New York (1985)zbMATHGoogle Scholar
  7. 7.
    Curtis, C.W.: On Lusztig’s isomorphism theorem for Hecke algebras. J. Algebra 92, 348–365 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garsia, A.M., Stanton, D.: Group actions on Stanley–Reisner rings and invariants of permutation groups. Adv. Math. 51(2), 107–201 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gnedin, A., Olshanski, G.: A q-analogue of de Finetti’s theorem. Electron. J. Comb. 16(1), 78 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gnedin, A., Olshanski, G.: q-Exchangeability via quasi-invariance. Ann. Probab. 38(6), 2103–2135 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jones, A.: A combinatorial approach to the double cosets of the symmetric group with respect to young subgroups. Eur. J. Comb. 17(7), 647–655 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kuan, J.: A Multi-species ASEP(q,j) and q-TAZRP with stochastic duality. Int. Mat. Res. Not. arXiv:1605.00691v1
  13. 13.
    Kuan, J.: An algebraic construction of duality functions for the stochastic \(U_q(A_n^{(1)})\) vertex model and its degenerations. Commun. Math. Phys. arXiv:1701.04468v2
  14. 14.
    Korhonen, M., Lee, E.: The transition probability and the probability for the left-most particle’s position of the q–TAZRP. J. Math. Phys. 55, 013301 (2014). arXiv:1308.4769v2 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976).  https://doi.org/10.1214/aop/1176996084 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sasamoto, T., Wadati, M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A. 31(28), 6057–6071 (1998).  https://doi.org/10.1088/0305-4470/31/28/019 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41, 255–268 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970).  https://doi.org/10.1016/0001-8708(70)90034-4 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Takeyama, Y.: Algebraic construction of multi-species q-Boson system. arXiv:1507.02033
  20. 20.
    Tits, J.: Groupes et géométries de Coxeter, mimeographed notes, Institut des Hautes Études Scientifiques (1961)Google Scholar
  21. 21.
    Tits, J.: Buildings of spherical type and finite BN-Pairs. Lecture Notes in Mathematics, No. 386, Springer, Berlin (1974)Google Scholar
  22. 22.
    Tracy, C., Widom, H.: Integral formulas for the asymmetric simple exclusion process. arXiv:0704.2633
  23. 23.
    Tracy, C., Widom, H.: On the distribution of a second class particle in the asymmetric simple exclusion process. arXiv:0907.4395
  24. 24.
    Tracy, C., Widom, H.: On the asymmetric simple exclusion process with multiple species, arXiv:1105.4906
  25. 25.
    Wang, D., Waugh, D.: The transition probability of the q-TAZRP (q-Bosons) with inhomogeneous jump rates. SIGMA 12, 037 (2016). arXiv:1512.01612v5

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College StationUSA

Personalised recommendations