Higher spin six vertex model and symmetric rational functions

Abstract

We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1\(+\)1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the inhomogeneous higher spin six vertex model for suitable domains. In the homogeneous case, such functions were previously studied in Borodin (On a family of symmetric rational functions, 2014); they also generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six vertex model.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Baxter, R.: Exactly Solved Models in Statistical Mechanics. Courier Dover Publications, New York (2007)

    Google Scholar 

  2. 2.

    Borodin, A., Bufetov, A.: An irreversible local Markov chain that preserves the six vertex model on a torus. Ann. Inst. Henri Poincare (to appear) (2015). arXiv:1509.05070 [math-ph]

  3. 3.

    Bogoliubov, N., Bullough, R., Timonen, J.: Critical behavior for correlated strongly coupled boson systems in \(1+1\) dimensions. Phys. Rev. Lett. 72(25), 3933–3936 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Borodin, A., Corwin, I.: Macdonald processes. Prob. Theory Rel. Fields 158, 225–400 (2014). arXiv:1111.4408 [math.PR]

    Article  Google Scholar 

  5. 5.

    Borodin, A., Corwin, I.: Discrete time q-TASEPs. Intern. Math. Res. Not. 2015(2), 499–537 (2015). arXiv:1305.2972 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Barraquand, G., Corwin, I.: Random-walk in Beta-distributed random environment. Probab. Theory Relat, Fields (2016). arXiv:1503.04117 [math.PR]

    Google Scholar 

  7. 7.

    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke J. Math. 165(3), 563–624 (2016). arXiv:1407.6729 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes. Trans. Am. Math. Soc. 368(3), 1517–1558 (2016). arXiv:1306.0659 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz. Comm. Math. Phys. 339(3), 1167–1245 (2015). arXiv:1407.8534 [math-ph]

  10. 10.

    Borodin, A., Corwin, I., Petrov, L., Sasamoto, T.: Spectral theory for the q-Boson particle system. Compos. Math. 151(1), 1–67 (2015). arXiv:1308.3475 [math-ph]

  11. 11.

    Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014). arXiv:1207.5035 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999). arXiv:math/9810105 [math.CO]

  13. 13.

    Bethe, H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. (On the theory of metals. I. Eigenvalues and eigenfunctions of the linear atom chain). Zeitschrift fur Physik 71, 205–226 (1931)

    Article  MATH  Google Scholar 

  14. 14.

    Borodin, A., Ferrari, P.: Anisotropic growth of random surfaces in \(2+1\) dimensions. Comm. Math. Phys. 325, 603–684 (2014). arXiv:0804.3035 [math-ph]

  15. 15.

    Bogoliubov, N., Izergin, A., Kitanine, N.: Correlation functions for a strongly correlated boson system. Nucl. Phys. B 516(3), 501–528 (1998). arXiv:solv-int/9710002

  16. 16.

    Borodin, A.: Schur dynamics of the Schur processes. Adv. Math. 228(4), 2268–2291 (2011). arXiv:1001.3442 [math.CO]

  17. 17.

    Borodin, A.: On a family of symmetric rational functions. arXiv preprint (2014). arXiv:1410.0976 [math.CO]

  18. 18.

    Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. 300, 71–155 (2016). arXiv:1305.5501 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Euro. Phys. Lett. 90(2), 20002 (2010)

    Article  Google Scholar 

  20. 20.

    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1 (2012). arXiv:1106.1596 [math.PR]

  21. 21.

    Corwin, I.: The \(q\)-Hahn Boson process and \(q\)-Hahn TASEP. Intern. Math. Res. Not. (rnu094) (2014). arXiv:1401.3321 [math.PR]

  22. 22.

    Corwin, I., Petrov, L.: The q-PushASEP: a new integrable model for traffic in 1+1 dimension. J. Stat. Phys. 160(4), 1005–1026 (2015). arXiv:1308.3124 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. Comm. Math. Phys. 343(2), 651–700 (2016). arXiv:1502.07374 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Dotsenko, V.: Replica Bethe ansatz derivation of the Tracy–Widom distribution of the free energy fluctuations in one-dimensional directed polymers. J. Stat. Mech.: Theory Exp. (07), P07010 (2010). arXiv:1004.4455 [cond-mat.dis-nn]

  25. 25.

    Fulman, J.: Probabilistic Measures and Algorithms Arising from the Macdonald Symmetric Functions. arxiv preprint (1997). arXiv:math/9712237 [math.CO]

  26. 26.

    Felder, G., Varchenko, A.: Algebraic Bethe ansatz for theelliptic quantum group \(E_{\tau ,\eta }({\rm sl}_2)\). Nucl.Phys. B 480(1–2), 485–503 (1996). arXiv:q-alg/9605024

  27. 27.

    Gnedin, A., Olshanski, G.: A q-analogue of de Finetti’s theorem. Electron. J. Comb. 16, R16 (2009). arXiv:0905.0367 [math.PR]

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Gnedin, A., Olshanski, G.: q-Exchangeability via quasi-invariance. Ann. Probab. 38(6), 2103–2135 (2010). arXiv:0907.3275 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Gwa, L.-H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68(6), 725–728 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6(3), 355–378 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Imamura, T., Sasamoto, T.: Current moments of 1D ASEP by duality. J. Stat. Phys. 142(5), 919–930 (2011). arXiv:1011.4588 [cond-mat.stat-mech]

  32. 32.

    Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153(1), 259–296 (2001). arXiv:math/9906120 [math.CO]

  33. 33.

    Korepin, V., Bogoliubov, N., Izergin, A.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  34. 34.

    König, W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447 (2005). arXiv:math/0403090 [math.PR]

  35. 35.

    König, W., O’Connell, N., Roch, S.: Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7(5), 1–24 (2002)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Kardar, M., Parisi, G., Zhang, Y.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889 (1986)

    Article  MATH  Google Scholar 

  37. 37.

    Kirillov, A.N., Reshetikhin, N.: Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum. J. Phys. A 20(6), 1565–1585 (1987)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Kulish, P., Reshetikhin, N., Sklyanin, E.: Yang-Baxter equation and representation theory: I. Lett. Math. Phys. 5(5), 393–403 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Koekoek, R., Swarttouw, R.F.: The Askey-Scheme of Hypergeometric Orthogonal Polynomials and Its q-Analogue. Technical report, Delft University of Technology and Free University of Amsterdam (1996)

  40. 40.

    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)

    Google Scholar 

  41. 41.

    Mangazeev, V.: On the Yang–Baxter equation for the six-vertex model. Nucl. Phys. B 882, 70–96 (2014). arXiv:1401.6494 [math-ph]

  42. 42.

    Matveev, K., Petrov, L.: \(q\)-randomized Robinson–Schensted–Knuth correspondences and random polymers. Ann. Inst. Henri Poincare (to appear) (2015). arXiv preprint arXiv:1504.00666 [math.PR]

  43. 43.

    O’Connell, N.: A path-transformation for random walks and the Robinson-Schensted correspondence. Trans. Am. Math. Soc. 355(9), 3669–3697 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    O’Connell, N.: Conditioned random walks and the RSK correspondence. J. Phys. A 36(12), 3049–3066 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    O’Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40(2), 437–458 (2012). arXiv:0910.0069 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  46. 46.

    Okounkov, A.: Infinite wedge and random partitions. Sel. Math. New Ser. 7(1), 57–81 (2001). arXiv:math/9907127 [math.RT]

  47. 47.

    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(3), 581–603 (2003). arXiv:math/0107056 [math.CO]

  48. 48.

    Povolotsky, A.: On integrability of zero-range chipping models with factorized steady state. J. Phys. A 46, 465205 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Reshetikhin, N.: Lectures on the integrability of the 6-vertex model. In: Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, pp. 197–266. Oxford University Press, Oxford (2010). arXiv:1010.5031 [math-ph]

  50. 50.

    Schütz, G.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86(5–6), 1265–1287 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  51. 51.

    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5(2), 246–290 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Sasamoto, T., Wadati, M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31, 6057–6071 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  53. 53.

    Tracy, C., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008). arXiv:0704.2633 [math.PR]. Erratum: Commun. Math. Phys. 304, 875–878 (2011)

  54. 54.

    Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290, 129–154 (2009). arXiv:0807.1713 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  55. 55.

    Tracy, C., Widom, H.: On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137, 825–838 (2009). arXiv:0907.5192 [math.PR]

    MathSciNet  Article  MATH  Google Scholar 

  56. 56.

    Vuletic, M.: Shifted Schur process and asymptotics of large random strict plane partitions. Int. Math. Res. Not. 2007(rnm043) (2007). arXiv:math-ph/0702068

  57. 57.

    Wang, D., Waugh, D.: The transition probability of the q-TAZRP (q-Bosons) with inhomogeneous jump rates. SIGMA Symmetry Integr. Geom: Methods Appl. 12(037) (2016). arXiv:1512.01612 [math.PR]

  58. 58.

    Wheeler, M., Zinn-Justin, P.: Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons. Adv. Math. 299, 543–600 (2016). arXiv:1508.02236 [math-ph]

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Leonid Petrov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Borodin, A., Petrov, L. Higher spin six vertex model and symmetric rational functions. Sel. Math. New Ser. 24, 751–874 (2018). https://doi.org/10.1007/s00029-016-0301-7

Download citation

Mathematics Subject Classification

  • Primary 60K35
  • Secondary 05E05
  • 82B23