Papers
Topics
Authors
Recent
Search
2000 character limit reached

Periodic $q$-Whittaker and Hall-Littlewood processes

Published 5 Oct 2023 in math.PR, math-ph, math.CO, and math.MP | (2310.03527v1)

Abstract: We study the periodic $q$-Whittaker and Hall-Littlewood processes, two probability measures on sequences of partitions. We prove that a certain observable of the periodic $q$-Whittaker process exhibits a $(q,u)$ symmetry after a random shift, generalizing a previous result of Imamura, Mucciconi, and Sasamoto who showed a matching between the periodic Schur and $q$-Whittaker measures, and also give a vertex model formulation of their result. As part of our proof of the $(q,u)$ symmetry, we obtain contour integral formulas for both the periodic $q$-Whittaker and Hall-Littlewood processes. We also show a matching between certain observables in the periodic Hall-Littlewood process and in a quasi-periodic stochastic six vertex model after a suitable random shift, and discuss a limit to the stationary periodic stochastic six vertex model.

Authors (2)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (26)
  1. Colored fermionic vertex models and symmetric functions, 2021. arXiv:2101.01605.
  2. J. Baik and Z. Liu. TASEP on a ring in sub-relaxation time scale. J. Stat. Phys., 165(6):1051–1085, 2016. doi:10.1007/s10955-016-1665-y.
  3. J. Baik and Z. Liu. Multipoint distribution of periodic TASEP. J. Amer. Math. Soc., 32(3):609–674, 2019. doi:10.1090/jams/915.
  4. J. Baik and Z. Liu. Periodic TASEP with general initial conditions. Probab. Theory Related Fields, 179(3-4):1047–1144, 2021. doi:10.1007/s00440-020-01004-6.
  5. Half-space Macdonald processes. Forum Math. Pi, 8:e11, 150, 2020. doi:10.1017/fmp.2020.3.
  6. Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process. Duke Math. J., 167(13):2457–2529, 2018. doi:10.1215/00127094-2018-0019.
  7. The free boundary Schur process and applications I. Ann. Henri Poincaré, 19(12):3663–3742, 2018. doi:10.1007/s00023-018-0723-1.
  8. A. Borodin. Periodic Schur process and cylindric partitions. Duke Math. J., 140(3):391–468, 2007. doi:10.1215/S0012-7094-07-14031-6.
  9. Between the stochastic six vertex model and Hall-Littlewood processes, 2016. arXiv:1611.09486.
  10. A. Borodin and I. Corwin. Macdonald processes. Probab. Theory Related Fields, 158(1-2):225–400, 2014. doi:10.1007/s00440-013-0482-3.
  11. Free energy fluctuations for directed polymers in random media in 1+1111+11 + 1 dimension. Comm. Pure Appl. Math., 67(7):1129–1214, 2014. doi:10.1002/cpa.21520.
  12. From duality to determinants for q𝑞qitalic_q-TASEP and ASEP. Ann. Probab., 42(6):2314–2382, 2014. doi:10.1214/13-AOP868.
  13. A. Borodin and M. Wheeler. Spin q𝑞qitalic_q-Whittaker polynomials. Adv. Math., 376:Paper No. 107449, 50, 2021. doi:10.1016/j.aim.2020.107449.
  14. A. Borodin and M. Wheeler. Nonsymmetric Macdonald polynomials via integrable vertex models. Trans. Amer. Math. Soc., 375(12):8353–8397, 2022. doi:10.1090/tran/8309.
  15. Matrix product formula for Macdonald polynomials. J. Phys. A, 48(38):384001, 25, 2015. doi:10.1088/1751-8113/48/38/384001.
  16. I. Corwin and E. Dimitrov. Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles. Comm. Math. Phys., 363(2):435–501, 2018. doi:10.1007/s00220-018-3139-3.
  17. Fluctuation exponents of the KPZ equation on a large torus. Comm. Pure Appl. Math., 76(11):3104–3149, 2023. doi:10.1002/cpa.22110.
  18. A. Garbali and M. Wheeler. Modified Macdonald polynomials and integrability. Comm. Math. Phys., 374(3):1809–1876, 2020. doi:10.1007/s00220-020-03680-w.
  19. Y. Gu and T. Komorowski. Fluctuations of the winding number of a directed polymer on a cylinder, 2023. arXiv:2207.14091.
  20. Y. Gu and T. Komorowski. KPZ on torus: Gaussian fluctuations, 2023. arXiv:2104.13540.
  21. Skew RSK dynamics: Greene invariants, affine crystals and applications to q𝑞qitalic_q-Whittaker polynomials, 2021. doi:10.48550/ARXIV.2106.11922.
  22. Solvable models in the KPZ class: approach through periodic and free boundary Schur measures, 2022. doi:10.48550/ARXIV.2204.08420.
  23. S. Koshida. Free field theory and observables of periodic Macdonald processes. J. Combin. Theory Ser. A, 182:Paper No. 105473, 42, 2021. doi:10.1016/j.jcta.2021.105473.
  24. Z. Liu. Height fluctuations of stationary TASEP on a ring in relaxation time scale. Ann. Inst. Henri Poincaré Probab. Stat., 54(2):1031–1057, 2018. doi:10.1214/17-AIHP831.
  25. I. G. Macdonald. Symmetric functions and Hall polynomials. The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs.
  26. V. Venkateswaran. Vanishing integrals for Hall–Littlewood polynomials. Transformation Groups, 17:259–302, 2012. doi:10.1007/s00031-012-9175-8.
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up