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Selecta Mathematica

, Volume 24, Issue 5, pp 4839–4884 | Cite as

Hall–Littlewood RSK field

  • Alexey Bufetov
  • Konstantin Matveev
Article
  • 7 Downloads

Abstract

We introduce a randomized Hall–Littlewood RSK algorithm and study its combinatorial and probabilistic properties. On the probabilistic side, a new model—the Hall–Littlewood RSK field—is introduced. Its various degenerations contain known objects (the stochastic six vertex model, the asymmetric simple exclusion process) as well as a variety of new ones. We provide formulas for a rich class of observables of these models, extending existing results about Macdonald processes. On the combinatorial side, we establish analogs of properties of the classical RSK algorithm: invertibility, symmetry, and a “bijectivization” of the skew-Cauchy identity.

Keywords

Hall–Littlewood polynomials RSK algorithm Macdonald processes Six vertex model 

Mathematics Subject Classification

Primary 05E05 Secondary 60K35 

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Notes

Acknowledgements

We are grateful to A. Borodin for useful discussions. We are grateful to I. Corwin for useful comments. We are grateful to referees for their useful remarks. A. Bufetov was partially supported by The Foundation Sciences Mathematiques de Paris.

References

  1. 1.
    Aggarwal, A.: Convergence of the stochastic six-vertex model to ASEP, preprint. arXiv:1607.08683
  2. 2.
    Borodin, A.: Stochastic higher spin six vertex model and Macdonald measures, preprint. arXiv:1608.01553
  3. 3.
    Borodin, A., Bufetov, A.: An irreversible local Markov chain that preserves the Six Vertex Model on a torus. Annales de l’Institut Henri Poincare Probability and Statistics, 53(1), 451–463 (2017). arXiv:1509.05070 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borodin, A., Bufetov, A., Wheeler, M.: Between the stochastix six vertex model and Hall–Littlewood processes. To appear in J. Comb. Theory Ser. A. arXiv:1611.09486
  5. 5.
    Borodin, A., Corwin, I.: Macdonald Processes. Probab. Theory Relat. Fields 158(1–2), 225–400 (2014). arXiv:1111.4408 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in 1 + 1 dimension. Commun. Pur. Appl. Math. 67(7), 1129–1214 (2014). arXiv:1204.1024 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke Math. J. 165(3), 563–624 (2016). arXiv:1407.6729 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes. Trans. Am. Math. Soc. 368, 1517–1558 (2016). arxiv:1306.0659 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Borodin, A., Corwin, I., Remenik, D.: Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity. Commun. Math. Phys. 324, 215 (2013). arXiv:1206.4573 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Borodin, A., Gorin, V.: General beta Jacobi corners process and the Gaussian free field. Commun. Pure Appl. Math. 68(10), 1774–1844 (2015). arXiv:1305.3627 CrossRefGoogle Scholar
  12. 12.
    Borodin, A., Olshanski, G.: Stochastic dynamics related to Plancherel measure on partitions. In: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics (V.Kaimanovich and A.Lodkin, eds). Amer. Math. Soc., Translations, Series 2, vol. 217, 2006, pp. 9–22. arXiv:math-ph/0402064
  13. 13.
    Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. 300, 71–155 (2016). arXiv:1305.5501 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Borodin, A., Petrov, L.: Integrable probability: From representation theory to Macdonald processes. Proba. Surv. 11, 1–58 (2014). arXiv:1310.8007 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. To appear in Selecta Math. arXiv:1601.05770
  16. 16.
    Bufetov, A., Knizel, A.: Asymptotics of random domino tilings of rectangular Aztec diamonds, preprint. arXiv:1604.01491
  17. 17.
    Bufetov, A., Petrov, L.: Law of large numbers for infinite random matrices over a finite field. Selecta Math. 21(4), 1271–1338 (2015). arXiv:1402.1772 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Bufetov, A., Petrov, L.: Yang–Baxter field for spin Hall–Littlewood symmetric functions, preprint. arXiv:1712.04584
  19. 19.
    Corwin, I.: Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class. Proceedings of the International Congress of Mathematicians (2014)Google Scholar
  20. 20.
    Corwin, I., Dimitrov, E.: Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall–Littlewood Gibbsian line ensembles, preprint. arXiv:1703.07180
  21. 21.
    Dimitrov, E.: KPZ and Airy limits of Hall–Littlewood random plane partitions, preprint. arXiv:1602.00727
  22. 22.
    Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S.: A commutative algebra on degenerate CP1 and Macdonald polynomials. J. Math. Phys. 50, 095215 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fomin, S.: Robinson–Schensted–Knuth, Generalized, correspondence, (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155. Translation in. J. Soviet Math. 41(1988), 979–991 (1986)Google Scholar
  24. 24.
    Fomin, S.: Schur operators and Knuth correspondences. J. Comb. Theory Ser. A 72(2), 277–292 (1995)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ghosal, P.: Hall–Littlewood-PushTASEP and its KPZ limit, preprint. arXiv:1701.07308
  26. 26.
    Gwa, L.H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68(6), 725–728 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Jochush, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem, Preprint (1995). arXiv:math/9801068
  28. 28.
    Johansson, K.: The arctic circle boundary and the Airy process. Ann. Probab. 33(1):1–30 (2005). arXiv:math/0306216 MathSciNetCrossRefGoogle Scholar
  29. 29.
    Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields, 123, 225–280 (2002). arXiv:math/0306216 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Knuth, D.: Permutations, matrices, and generalized Young tableaux. Pac. J. Math. 34(3), 709–727 (1970)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1999)Google Scholar
  33. 33.
    Matveev, K., Petrov, L.: q-randomized Robinson–Schensted–Knuth correspondences and random polymers (2015). Annales de l’Institut Henri Poincar’e D: Combinatorics, Physics and their Interactions 4(1), 1–123 (2017). arXiv:1504.00666
  34. 34.
    O’Connell, N., Pei, Y.: A q-weighted version of the Robinson–Schensted algorithm. Electron. J. Probab. 18(95), 1–25 (2013). arXiv:1212.6716 MathSciNetzbMATHGoogle Scholar
  35. 35.
    O’Connell, N., Warren, J.: A multi-layer extension of the stochastic heat equation. Commun. Math. Phys. 341(1), 1–33 (2016). arXiv:1104.3509 MathSciNetCrossRefGoogle Scholar
  36. 36.
    Okounkov, A.: Infinite wedge and random partitions. Selecta Math. 7, 51–81 (2001)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581–603 (2003)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Pei, Y.: A q-Robinson–Schensted–Knuth algorithm and a q-polymer, preprint. arXiv:1610.03692
  39. 39.
    Prahofer, M., Spohn, H.: Scale invariance of the PNG droplet and the airy process. J. Stat. Phys. 108, 1071 (2002). arXiv:math/0105240 MathSciNetCrossRefGoogle Scholar
  40. 40.
    de B. Robinson, G.: On the resentations of \(S_n\). Am. J. Math. 60, 745–760 (1938)Google Scholar
  41. 41.
    Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 12, 117–128 (1963)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Shiraishi, J.: A family of integral transformations and basic hypergeometric series. Commun. Math. Phys 263, 439–460 (2006)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Stanley, R.: Enumerative Combinatorics, Vol. 2. Cambridge Studies in Advanced Mathematics 62. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  44. 44.
    Tracy, C.A., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291–300 (2008)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290(1), 129–154 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Tracy, C.A., Widom, H.: On the asymmetric simple exclusion process with multiple species. J. Stat. Phys. 150, 457–470 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsBrandeis UniversityWalthamUSA

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