Journal of Statistical Physics

, Volume 174, Issue 1, pp 1–27 | Cite as

Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

  • F. Colomo
  • A. G. PronkoEmail author
  • A. Sportiello


We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of free-fermion vertex weights. We describe how the recently developed ‘Tangent method’ can be used to determine the form of the arctic curve. The obtained result is in agreement with numerics.


Vertex models Domain wall boundary conditions Limit shape phenomena Arctic curves Tangent method Log-gas representations 



We are grateful to A. Abanov, S. Chhita and F. Franchini for interesting discussions. We are indebted to B. Wieland for sharing with us the code for generating uniformly sampled alternating-sign matrices. We thank the Simons Center for Geometry and Physics (SCGP, Stony Brook), research program on ‘Statistical Mechanics and Combinatorics’ and the Galileo Galilei Institute for Theoretical Physics (GGI, Florence), research programs on ‘Statistical Mechanics, Integrability and Combinatorics’ and ‘Entanglement in Quantum Systems’, for hospitality and support at some stage of this work. FC is grateful to LIPN, équipe Calin at Université Paris 13, for hospitality and support at some stage of this work. AGP and AS are grateful to INFN, Sezione di Firenze for hospitality and support at some stage of this work. AGP acknowledges partial support from the Russian Science Foundation, Grant #18-11-00297.


  1. 1.
    Korepin, V.E., Zinn-Justin, P.: Thermodynamic limit of the six-vertex model with domain wall boundary conditions. J. Phys. A 33, 7053–7066 (2000). arXiv:cond-mat/0004250 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Zinn-Justin, P.: Six-vertex model with domain wall boundary conditions and one-matrix model. Phys. Rev. E 62, 3411–3418 (2000). arXiv:math-ph/0005008 ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Zinn-Justin, P.: The influence of boundary conditions in the six-vertex model (2002), arXiv:cond-mat/0205192
  4. 4.
    Reshetikhin, N., Palamarchuk, K.: The 6-vertex model with fixed boundary conditions, (2006) arXiv:1010.5011
  5. 5.
    Colomo, F., Pronko, A.G.: The arctic curve of the domain-wall six-vertex model. J. Stat. Phys. 138, 662–700 (2010). arXiv:0907.1264 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bleher, P., Liechty, K.: Random matrices and the six-vertex model. In: CRM Monographs Series, vol. 32. American Mathematical Society, Providence, RI (2013)Google Scholar
  7. 7.
    Reshetikhin, N., Sridhar, A.: Integrability of limit shapes of the six-vertex model. Commun. Math. Phys. 356, 535–563 (2017). arXiv:1510.01053 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Reshetikhin, N., Sridhar, A.: Limit shapes of the stochastic six-vertex model (2016), arXiv:1609.01756
  9. 9.
    Allegra, N., Dubail, J., Stéphan, J.-M., Viti, J.: Inhomogeneous field theory inside the arctic circle. J. Stat. Mech. 2016, 053108 (2016). arXiv:1512.02872 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borodin, A., Corwin, I., Gorin, V.: Stochastic six-vertex model. Duke Math. J. 165, 563–624 (2016). arXiv:1407.6729 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dimitrov, E.: Six-vertex models and the GUE-corners process. Int. Math. Res. Notices (2018), in press arXiv:1610.06893
  12. 12.
    Granet, A., Budzynzki, L., Dubail, J., Jacobsen, J.L.: Inhomogeneous Gaussian free field inside the interacting Arctic curve, arXiv:1807.07927
  13. 13.
    Jockush, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem, arXiv:math/9801068
  14. 14.
    Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. N. Y. J. Math. 4, 137–165 (1998). arXiv:math/9801059 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the \(3D\) Ising model. Commun. Math. Phys. 222, 147–179 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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). arXiv:math/0107056 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1–46 (2003). arXiv:cond-mat/0212456 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kenyon, R., Okounkov, A.: Planar dimers and Harnack curves. Duke Math. J. 131, 499–524 (2006). arXiv:math-ph/0311062 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006). arXiv:math-ph/0311005 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta Math. 199, 263–302 (2007). arXiv:math-ph/0507007 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Petersen, T.K., Speyer, D.: An arctic circle theorem for groves. J. Comb. Theory. Ser. A 111, 137–164 (2005). arXiv:math/0407171 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pittel, B., Romik, D.: Limit shapes for random square Young tableaux. Adv. Appl. Math. 38, 164–209 (2007). arXiv:math.PR/0405190 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Francesco, P.Di, Soto-Garrido, R.: Arctic curves of the octahedron equation. J. Phys. A 47, 285204 (2014). arXiv:1402.4493 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Petrov, L.: Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes. Prob. Theor. Rel. Fields 160, 429–487 (2014). arXiv:1202.3901 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Romik, D., Śniady, P.: Limit shapes of bumping routes in the Robinson-Schensted correspondence. Random Struct. Algorithm 48, 171–182 (2016). arXiv:1304.7589 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Boutillier, C., Bouttier, J., Chapuy, G., Corteel, S., Ramassamy, S.: Dimers on rail yard graphs. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4, 479–539 (2017). arXiv:1504.05176 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Bufetov, A., Knizel, A.: Asymptotics of random domino tilings of rectangular Aztec diamonds. Ann. Inst. H. Poincar Probab. Statist. 54, 1250–1290 (2018). arXiv:1604.01491 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Francesco, P.Di, Lapa, M.F.: Arctic curves in path models from the tangent method. J. Phys. A 51, 155202 (2018). arXiv:1711.03182 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Francesco, P. Di, Guitter, E.: Arctic curves for paths with arbitrary starting points: a tangent method approach. J. Phys. A: Math. Theor. (2018) arXiv:1803.11463. In press
  30. 30.
    Stéphan, J.-M.: Return probability after a quantum quench from a domain wall initial state in the spin-1/2 XXZ chain. J. Stat. Mech. Theory Exp. 2017, 103108 (2017). arXiv:1707.06625 CrossRefGoogle Scholar
  31. 31.
    Collura, M., Luca, A.De, Viti, J.: Analytic solution of the domain wall nonequilibrium stationary state. Phys. Rev. B 97, 081111 (2018). arXiv:1707.06218 ADSCrossRefGoogle Scholar
  32. 32.
    Cugliandolo, L.: Artificial spin-ice and vertex models. J. Stat. Phys. 167, 499–514 (2017). arXiv:1701.02283 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Korepin, V.E.: Calculations of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Colomo, F., Pronko, A.G.: The limit shape of large alternating-sign matrices. SIAM J. Discret. Math. 24, 1558–1571 (2010). arXiv:0803.2697 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Colomo, F., Pronko, A. G., Zinn-Justin, P.: The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime. J. Stat. Mech. Theory Exp. L03002 (2010) arXiv:1001.2189
  36. 36.
    Colomo, F., Sportiello, A.: Arctic curves of the six-vertex model on generic domains: the tangent method. J. Stat. Phys. 164, 1488–1523 (2016). arXiv:1605.01388 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Colomo, F., Pronko, A.G.: Emptiness formation probability in the domain-wall six-vertex model. Nucl. Phys. B 798, 340–362 (2008). arXiv:0712.1524 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000). arXiv:math/9903134 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pronko, A.G.: On the emptiness formation probability in the free-fermion six-vertex model with domain wall boundary conditions. J. Math. Sci. (N. Y.) 192, 101–116 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Colomo, F., Pronko, A.G.: Third-order phase transition in random tilings. Phys. Rev. E 88, 042125 (2013). arXiv:1306.6207 ADSCrossRefGoogle Scholar
  41. 41.
    Colomo, F., Pronko, A.G.: Thermodynamics of the six-vertex model in an L-shaped domain. Commun. Math. Phys. 339, 699–728 (2015). arXiv:1501.03135 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Colomo, F., Pronko, A.G., Sportiello, A.: Generalized emptiness formation probability in the six-vertex model. J. Phys. A 49, 415203 (2016). arXiv:1605.01700 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating-sign matrices and domino tilings. J. Algebraic Combin. 1, 111– 132; 219– 234, (1992) arXiv:math/9201305
  44. 44.
    Propp, J.: Generalized domino-shuffling. Theor. Comput. Sci. 303, 267–301 (2003). arXiv:math/0111034 MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Baik, J., Kriecherbauer, T., McLaughlin, K.T.-R., Miller, P.D.: Discrete orthogonal polynomials: asymptotics and applications. In: Ann. of Math. Stud., vol. 164. Princeton University Press, Princeton, NJ (2007)Google Scholar
  46. 46.
    Douglas, M.R., Kazakov, V.A.: Large \(N\) phase transition in continuum \(\text{ QCD }_2\). Phys. Lett. B 319, 219–230 (1993). arXiv:hep-th/9305047 ADSCrossRefGoogle Scholar
  47. 47.
    Zinn-Justin, P.: Universality of correlation functions of Hermitian random matrices in an external field. Commun. Math. Phys. 194, 631–650 (1998). arXiv:cond-mat/9705044 ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.INFN, Sezione di FirenzeSesto FiorentinoItaly
  2. 2.Steklov Mathematical InstituteSt. PetersburgRussia
  3. 3.LIPN, and CNRS, Université Paris 13, Sorbonne Paris CitéVilletaneuseFrance

Personalised recommendations