Skip to main content
SpringerLink
Log in

Thermodynamics of the Six-Vertex Model in an L-Shaped Domain

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions. For free-fermion vertex weights the partition function can be expressed in terms of some Hankel determinant, or equivalently as a Coulomb gas with discrete measure and a non-polynomial potential with two hard walls. We use Coulomb gas methods to study the partition function in the thermodynamic limit. We obtain the free energy of the six-vertex model as a function of the parameters describing the geometry of the scaled L-shaped domain. Under variations of these parameters the system undergoes a third-order phase transition. The result can also be considered in the context of dimer models, for the perfect matchings of the Aztec diamond graph with a cut-off corner.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Grensing D., Carlsen I., Zapp H.C.: Some exact results for the dimer problem on plane lattices with non-standard boundaries. Phil. Mag. A 41, 777–781 (1980)

    Article  ADS  Google Scholar 

  2. 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

  3. Jockusch, W., Propp, J., Shor, P.: Random domino tilings and the arctic circle theorem. arXiv:math/9801068

  4. Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. New York J. Math. 4, 137–165 (1998). arXiv:math/9801059

  5. Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta Math., 199, 263–302 (2007). arXiv:math-ph/0507007

  6. Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. of Math., 163, 1019–1056 (2006). arXiv:math-ph/0311005

  7. Kenyon, R., Okounkov, A.: Planar dimers and Harnack curves. Duke Math. J. 131, 499–524 (2006). arXiv:math-ph/0311005

  8. Kenyon, R.: Lectures on dimers, IAS/Park City Math. Ser, 16, 191–230 (2009). arXiv:0910.3129

  9. 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

  10. Reshetikhin, N., Palamarchuk, K.: The 6-vertex model with fixed boundary conditions. PoS, Solvay, 012 (2006). arXiv:cond-mat/0502314

  11. 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

  12. Colomo, F., Pronko, A.G.: Third-order phase transition in random tilings. Phys. Rev. E, 88, 042125 (2013). arXiv:1306.6207

  13. Johansson, K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209, 437–476 (2000). arXiv:math/9903134

  14. Johansson, K.: Non-intersecting paths, random tilings, and random matrices. Probab. Theory Related Fields, 123, 225–280 (2002). arXiv:math/0011250

  15. 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

  16. Bleher P., Liechty K.: Random matrices and the six vertex model, CRM Monograph Series, vol. 32. American Mathematical Society, Providence (2013)

    Google Scholar 

  17. Baik J., Kriecherbauer T., McLaughlin K.T.-R., Miller P.D.: Discrete orthogonal polinomials: Asymptotics and applications, Ann. of Math. Stud. vol. 164. Princeton University Press, Princeton (2007)

    Google Scholar 

  18. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, San Diego (1982)

    MATH  Google Scholar 

  19. Korepin V.E.: Calculations of norms of Bethe wave functions. Comm. Math. Phys. 86, 391–418 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Izergin A.G.: Partition function of the six-vertex model in the finite volume. Sov. Phys. Dokl. 32, 878–879 (1987)

    ADS  MATH  Google Scholar 

  21. Izergin A.G., Coker D.A., Korepin V.E.: Determinant formula for the six-vertex model. J. Phys. A 25, 4315–4334 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  22. Bogoliubov, N.M., Pronko, A.G., Zvonarev, M.B.: Boundary correlation functions of the six-vertex model. J. Phys. A, 35, 5525–5541 (2002). arXiv:math-ph/0203025

  23. 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

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. Forrester, P.J., Witte, N.S.: Application of the τ-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J., 174, 29–114 (2004). arXiv:math-ph/0204008

  26. Majumdar, S.N., Schehr, G.: Top eigenvalue of a random matrix: large deviations and third order phase transition. J. Stat. Mech. Theory Exp. (2014). arXiv:1311.0580

  27. Douglas, M.R., Kazakov, V.A.: Large N phase transition in continuum QCD2. Phys. Lett. B, 319, 219–230 (1993). arXiv:hep-th/9305047

  28. Tracy, C.A., Widom, H.: Level spacing distributions and the Airy kernel. Comm. Math. Phys., 159, 151–174 (1994). arXiv:hep-th/9211141

  29. Dragnev P.D., Saff E.B.: Constrained energy problems with applications to orthogonal polynomials of a discrete variable. J. Anal. Math. 72(1), 223–259 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kuijlaars A.B.J.: On the finite-gap ansatz in the continuum limit of the Toda lattice. Duke Math. J. 104(3), 433–462 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Koekoek R., Swarttouw R.F., Lesky P.A.: Hypergeometric orthogonal polynomials and their q-analogues. Springer Monographs in Mathematics, Springer-Verlag, Berlin (2010)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INFN Sezione di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, FI, Italy

    Filippo Colomo

  2. Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

    Andrei G. Pronko

Authors
  1. Filippo Colomo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrei G. Pronko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei G. Pronko.

Additional information

Communicated by H. Spohn

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Colomo, F., Pronko, A.G. Thermodynamics of the Six-Vertex Model in an L-Shaped Domain. Commun. Math. Phys. 339, 699–728 (2015). https://doi.org/10.1007/s00220-015-2406-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2406-9

Keywords

Search

Navigation