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Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model

  • Amol AggarwalEmail author
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Abstract

In this paper we consider the stochastic six-vertex model on a cylinder with arbitrary initial data. First, we show that it exhibits a limit shape in the thermodynamic limit, whose density profile is given by the entropy solution to an explicit, non-linear conservation law that was predicted by Gwa–Spohn (Phys Rev Lett 68:725–728, 1992) and by Reshetikhin–Sridhar (Commun Math Phys 363:741–765, 2018). Then, we show that the local statistics of this model around any continuity point of its limit shape are given by an infinite-volume, translation-invariant Gibbs measure of the appropriate slope.

Notes

Acknowledgements

The author heartily thanks Alexei Borodin, Ivan Corwin, and Jeffrey Kuan for enlightening discussions. The author is also grateful to the anonymous referee for helpful suggestions on an earlier draft of this manuscript. This work was partially supported by the NSF Graduate Research Fellowship under Grant Number DGE1144152 and NSF Grant DMS-1664619.

References

  1. 1.
    Aggarwal, A.: Convergence of the stochastic six-vertex model to the ASEP. Math. Phys. Anal. Geom. 20, 3 (2017)zbMATHCrossRefGoogle Scholar
  2. 2.
    Aggarwal, A.: Current fluctuations of the stationary ASEP and stochastic six-vertex model. Duke Math. J. 167, 269–384 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aggarwal, A., Borodin, A.: Phase transitions in the ASEP and stochastic six-vertex model. Ann. Prob. 47, 613–689 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aggarwal, A., Borodin, A., Bufetov, A.: Stochasticization of solutions to the Yang–Baxter equation. Ann. Henri Poincaré 20, 2495–2554 (2019)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \({\mathbb{Z}}\). J. Stat. Phys. 47, 265–288 (1987)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: A constructive approach to Euler hydrodynamics for attractive processes, application to \(k\)-step exclusion. Appl. Stoch. Process 99, 1–30 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Constructive Euler Hydrodynamics for One-Dimensional Attractive Particle Systems. In: Sidoravicius, V. (ed.) Sojourns in Probability Theory and Statistical Physics III, pp. 43–89. Springer, Singapore (2019)Google Scholar
  8. 8.
    Bahadoran, C., Mountford, T.S.: Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process. Probab. Theory Relat. Fields 136, 341–362 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Baik, J., Kriecherbauer, T., McLaughlin, K.T.-R., Miller, P.D.: Discrete Orthogonal Polynomials: Asymptotics and Applications, Ann. Math. Studies, Princeton Univ. Press (2007)Google Scholar
  10. 10.
    Barraquand, G., Borodin, A., Corwin, I., Wheeler, M.: Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process. Duke Math. J. 167, 2457–2529 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1989)zbMATHGoogle Scholar
  12. 12.
    Bazhanov, V.V.: Trigonometric solutions of triangle equations and classical Lie algebras. Phys. Lett. B 159, 321–324 (1985)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Benassi, A., Fouque, J.-P.: Hydrodynamical limit for the asymmetric simple exclusion process. Ann. Prob. 15, 546–560 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Borodin, A.: Stochastic higher spin six vertex model and Macdonald measures. J. Math. Phys. 59, 023301 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke Math. J. 165, 563–624 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Borodin, A., Gorin, V.: A Stochastic Telegraph Equation From the Six-Vertex Model, To appear in Ann. Prob., preprint, arXiv:1803.09137
  17. 17.
    Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. Sel. Math. 24, 751–874 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Borodin, A., Wheeler, M.: Coloured Stochastic Vertex Models and Their Spectral Theory, preprint, arXiv:1808.01866
  19. 19.
    Bukman, D.J., Shore, J.D.: The conical point in the ferroelectric six-vertex model. J. Stat. Phys. 78, 1277–1309 (1995)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Chhita, S., Johansson, K.: Domino statistics of the two-periodic Aztec diamond. Adv. Math. 294, 37–149 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chhita, S., Johansson, K., Young, B.: Asymptotic domino statistics in the Aztec diamond. Ann. Appl. Prob. 25, 1232–1278 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Cohn, H., Elkies, N., Propp, J.: Local Statistics of random domino tilings of the Aztec diamond. Duke Math. J. 85, 117–166 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Am. Math. Soc. 14, 297–346 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Corwin, I., Dimitrov, E.: Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall–Littlewood Gibbsian line ensembles. Commun. Math. Phys. 363, 435–501 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Corwin, I., Ghosal, P., Shen, H., Tsai, L.-C.: Stochastic PDE Limit of the Six Vertex Model, preprint, arXiv:1803.08120
  26. 26.
    Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343, 651–700 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Corwin, I., Tsai, L.-C.: KPZ equation limit of higher-spin exclusion processes. Ann. Prob. 45, 1771–1798 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Duits, M., Kuijlaars, M.B.J.: The Two Periodic Aztec Diamond and Matrix Valued Orthogonal Polynomials, To appear In: J. Eur. Math. Soc., preprint, arXiv:1712.05636
  30. 30.
    de Gier, J., Kenyon, R., Watson, S.S.: Limit Shapes for the Asymmetric Five Vertex Model, preprint, arXiv:1812.11934
  31. 31.
    Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697–715 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Gorin, V.: Bulk Universality for random Lozenge tilings near straight boundaries and for tensor products. Commun. Math. Phys. 354, 317–344 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Gorin, V.: Nonintersecting paths and the Hahn orthogonal ensemble. Funct. Anal. Appl. 42, 180–197 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Gwa, L.-H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian. Phys. Rev. Lett. 68, 725–728 (1992)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Prob. 6, 355–378 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Jayaprakash, C., Saam, W.F.: Thermal evolution of crystal shapes: the fcc crystal. Phys. Rev. B 30, 3916 (1984)ADSCrossRefGoogle Scholar
  37. 37.
    Jimbo, M.: Quantum \(R\) matrix for the generalized Toda system. Commun. Math. Phys. 102, 537–547 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Johansson, K.: The Arctic Circle boundary and the Airy process. Ann. Prob. 33, 1–30 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kasteleyn, P.W.: The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)ADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Kenyon, R.: Conformal invariance of domino tiling. Ann. Prob. 28, 759–795 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Kenyon, R.: Dominos and the Gaussian free field. Ann. Prob. 29, 1128–1137 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kenyon, R.: Lectures on Dimers, In: Statistical Mechanics, IAS/Park City Math. Ser. 16, Am. Math. Soc., Providence, RI, 191–230 (2009)Google Scholar
  43. 43.
    Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. Henri Poincaré Probab. Stat. 33, 591–618 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta Math. 199, 263–302 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and Amoebae. Ann. Math. 163, 1019–1056 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Kipnis, C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)zbMATHCrossRefGoogle Scholar
  47. 47.
    Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs in Mathematical Physics, Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar
  48. 48.
    Kosygina, E.: The behavior of the specific entropy in the hydrodynamical scaling limit. Ann. Prob. 29, 1086–1110 (2001)zbMATHCrossRefGoogle Scholar
  49. 49.
    Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sb. 10, 217–243 (1970)CrossRefGoogle Scholar
  50. 50.
    Kuan, J.: An algebraic construction of duality functions for the stochastic \({\cal{U}}_q ( A_n^{(1)})\) vertex model and its degenerations. Commun. Math. Phys. 359, 121–187 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Kuniba, A., Mangazeev, V.V., Maruyama, S., Okado, M.: Stochastic \(R\) matrix for \(U_q (A_n^{(1)})\). Nucl. Phys. B 913, 248–277 (2016)ADSzbMATHCrossRefGoogle Scholar
  52. 52.
    Laslier, B.: Local limits of Lozenge tilings are stable under bounded boundary height perturbations. Probab. Theory Relat. Fields 173, 1243–1264 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Lieb, E.H.: Residual entropy of square ice. Phys. Rev. Lett. 162, 162–172 (1967)ADSGoogle Scholar
  54. 54.
    Liggett, T.M.: Coupling the simple exclusion process. Ann. Prob. 3, 339–356 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Neergard, J., den Nijs, M.: Crossover scaling functions in one dimensional dynamic growth crystals. Phys. Rev. Lett. 74, 730 (1995)ADSCrossRefGoogle Scholar
  56. 56.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Palamarchuk, K., Reshetikhin, N.: The 6-vertex Model with Fixed Boundary Conditions, Proceedings of Solvay Workshop “Bethe Ansatz: 75 Years Later,” (2006)Google Scholar
  58. 58.
    Petrov, L.: Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes. Probab. Theory Relat. Fields 160, 429–487 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Reshetikhin, N., Sridhar, A.: Integrability of limit shapes of the six vertex model. Commun. Math. Phys. 356, 535–565 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Reshetikhin, N., Sridhar, A.: Limit shapes of the stochastic six-vertex model. Commun. Math. Phys. 363, 741–765 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on \({\mathbb{Z}}^d\). Commun. Math. Phys. 140, 417–448 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Russkikh, M.: Dimers in piecewise Temperleyan domains. Commun. Math. Phys. 359, 189–222 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Russkikh, M.: Dominos in Hedgehog Domains, To appear in Ann. Inst. Henri Poincaré D, preprint, arXiv:1803.10012
  64. 64.
    Seppäläinen, T.: Existence of hydrodynamics for the totally asymmetric simple \(K\)-exclusion process. Ann. Prob. 27, 361–415 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Serre, D.: Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves, Translated From the 1996 French Original by I. N. Sneddon, Cambridge University Press, Cambridge, (1999)Google Scholar
  66. 66.
    Sheffield, S.: Random Surfaces. Astérisque 304, (2005)Google Scholar
  67. 67.
    Shore, J., Bukman, D.J.: Coexistence point in the six-vertex model and the crystal shape of fcc materials. Phys. Rev. Lett. 72, 604–607 (1994)ADSCrossRefGoogle Scholar
  68. 68.
    Sutherland, B., Yang, C.N., Yang, C.P.: Exact solution of a model of two-dimensional ferroelectrics in an arbitrary external electric field. Phys. Rev. Lett. 19, 588–591 (1967)ADSCrossRefGoogle Scholar
  69. 69.
    Yau, H.-T.: Relative entropy and hydrodynamics of Ginzburg–Landau models. Lett. Math. Phys. 22, 63–80 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Zinn-Justin, P.: The Influence of Boundary Conditions in the Six-Vertex Model, preprint, arXiv:cond-mat/0205192v1

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Harvard UniversityCambridgeUSA

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