Higher spin six vertex model and symmetric rational functions



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.

Mathematics Subject Classification

Primary 60K35 Secondary 05E05 82B23 


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  1. 1.
    Baxter, R.: Exactly Solved Models in Statistical Mechanics. Courier Dover Publications, New York (2007)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Borodin, A., Corwin, I.: Macdonald processes. Prob. Theory Rel. Fields 158, 225–400 (2014). arXiv:1111.4408 [math.PR]MathSciNetCrossRefGoogle 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]MathSciNetMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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)CrossRefGoogle 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]MathSciNetCrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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]MathSciNetMATHGoogle Scholar
  28. 28.
    Gnedin, A., Olshanski, G.: q-Exchangeability via quasi-invariance. Ann. Probab. 38(6), 2103–2135 (2010). arXiv:0907.3275 [math.PR]MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6(3), 355–378 (1978)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  36. 36.
    Kardar, M., Parisi, G., Zhang, Y.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889 (1986)CrossRefMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Kulish, P., Reshetikhin, N., Sklyanin, E.: Yang-Baxter equation and representation theory: I. Lett. Math. Phys. 5(5), 393–403 (1981)MathSciNetCrossRefMATHGoogle 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)Google Scholar
  40. 40.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    O’Connell, N.: Conditioned random walks and the RSK correspondence. J. Phys. A 36(12), 3049–3066 (2003)MathSciNetCrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5(2), 246–290 (1970)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Sasamoto, T., Wadati, M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31, 6057–6071 (1998)MathSciNetCrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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]MathSciNetCrossRefMATHGoogle 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]

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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