Abstract
In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon, whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo and Sportiello (J Stat Phys 164:1488–1523, 2016) as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.
Similar content being viewed by others
Notes
Here, we always orient the edges of \({\mathbb {Z}}^2\) to the north or to the east; this makes \({\mathbb {Z}}^2\) into a directed graph.
This discrepancy arises from the fact that slightly different symmetries are required to transform the southeast part of the arctic boundary to its other parts in the cases of the three-bundle domain \({\mathcal {T}}\) and the square domain \({\mathcal {S}}\).
One might view \(-\infty \) and \(\infty \) as consisting of the unique vertices \(\big \{ (-\infty , -\infty ) \big \}\) and \(\big \{ (\infty , \infty ) \big \}\), respectively.
References
Allegra, N., Dubail, J., Stéphan, J.-M., Viti, J.: Inhomogeneous field theory inside the arctic circle. J. Stat. Mech. Theory Exp. 5, 053108 (2016)
Allison, D., Reshetikhin, N.: Numerical study of the 6-vertex model with domain wall boundary conditions. Ann. Inst. Fourier 55, 1847–1869 (2005)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1989)
Bleher, P.M., Fokin, V.V.: Exact solution of the six-vertex model with domain wall boundary conditions. Disordered phase. Commun. Math. Phys. 268, 223–284 (2006)
Borodin, A., Gorin, V., Rains, E.M.: \(q\)-Distributions on boxed plane partitions. Selecta Math. 16, 731–789 (2010)
Boutillier, C., Bouttier, J., Chapuy, G., Corteel, S., Ramassamy, S.: Dimers on rail yard graphs. Ann. Inst. Henri Poincaré D 4, 479–539 (2017)
Bufetov, A., Knizel, A.: Asymptotics of random domino tilings of rectangular Aztec diamonds. Ann. Inst. Henri Poincaré Probab. Stat. 54, 1250–1290 (2018)
Cantini, L., Sportiello, A.: A one-parameter refinement of the Razumov–Stroganov correspondence. J. Combin. Theory Ser. A 127, 400–440 (2014)
Cantini, L., Sportiello, A.: Proof of the Razumov–Stroganov conjecture. J. Combin. Theory Ser. A 118, 1549–1574 (2011)
Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Commun. Math. Phys. 222, 147–179 (2001)
Cohn, H., Elkies, N., Propp, J.: Local statistics of random domino tilings of the Aztec diamond. Duke Math. J. 85, 117–166 (1996)
Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Am. Math. Soc. 14, 297–346 (2001)
Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. N. Y. J. Math. 4, 137–165 (1998)
Colomo, F., Pronko, A.G.: Emptiness formation probability in the domain-wall six-vertex model. Nucl. Phys. B 798, 340–362 (2008)
Colomo, F., Pronko, A.G.: The arctic curve of the domain-wall six-vertex model. J. Stat. Phys. 138, 662–700 (2010)
Colomo, F., Pronko, A.G.: The arctic circle revisited. In: Baik, J., Kriecherbauer, T., Li, L.-C., McLaughlin, K.D.T.-R., Tomei, C. (eds.) Integrable Systems and Random Matrices, Contemporary Mathematics, vol. 458, pp. 361–376. American Mathematical Society, Providence (2008)
Colomo, F., Pronko, A.G.: The limit shape of large alternating sign matrices. SIAM J. Discrete Math. 24, 1558–1571 (2010)
Colomo, F., Pronko, A.G.: Thermodynamics of the six-vertex model in an L-shaped domain. Commun. Math. Phys. 339, 699–728 (2015)
Colomo, F., Pronko, A.G., Sportiello, A.: Arctic curves of the free-fermion six-vertex model in an L-shaped domain. J. Stat. Phys. 174, 1–27 (2019)
Colomo, F., Sportiello, A.: Arctic curves of the six-vertex model on generic domains: the tangent method. J. Stat. Phys. 164, 1488–1523 (2016)
Colomo, F., Sportiello, A.: In preparation
Corteel, S., Keating, D., Nicoletti, M.: Arctic Curves Phenomena for Bounded Lecture Hall Tableaux. arXiv:1905.02881
Corwin, I., Hammond, A.: Brownian Gibbs property for Airy line ensembles. Invent. Math. 195, 441–508 (2014)
Corwin, I., Hammond, A.: KPZ line ensemble. Probab. Theory Relat. Fields 166, 67–185 (2016)
Cugliando, L.F., Gonnella, G., Pelizzola, A.: Six-vertex model with domain wall boundary conditions in the Bethe–Peierls approximation. J. Stat. Mech. Theory Exp. 2015, P06008 (2015)
Debin, B., Di Francesco, P., Guitter, E.: Arctic Curves of the Twenty-Vertex Model with Domain Wall Boundaries. arXiv:1910.06833
Debin, B., Granet, E., Ruelle, P.: Concavity Analysis of the Tangent Method. arXiv:1905.11277
Debin, B., Ruelle, P.: Tangent Method for the Arctic Curve Arising From Freezing Boundaries. arXiv:1810.04909
Di Francesco, P., Guitter, E.: A tangent method derivation of the arctic curve for \(q\)-weighted paths with arbitrary starting points. J. Phys. A 52, 11 (2019)
Di Francesco, P., Guitter, E.: Arctic curves for paths with arbitrary starting points: a tangent method approach. J. Phys. A 51, 355201 (2018)
Di Francesco, P., Guitter, E.: The arctic curve for aztec rectangles with defects via the tangent method. J. Stat. Phys. 176, 639–678 (2019)
Di Francesco, P., Lapa, M.F.: Arctic curves from the tangent method. J. Phys. A 51, 155202 (2018)
Di Francesco, P., Soto-Garrido, R.: Arctic curves of the octahedron equation. J. Phys. A 47, 285204 (2014)
Eloranta, K.: Diamond ice. J. Stat. Phys. 96, 1091–1109 (1999)
George, T.: Grove Arctic Circles from Periodic Cluster Modular Transformations. arXiv:1711.00790
Gorin, V.: From alternating sign matrices to the Gaussian unitary ensemble. Commun. Math. Phys. 332, 437–447 (2014)
Gorin, V., Panova, G.: Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43, 3052–3132 (2015)
Izergin, A.G.: Partition function of the six-vertex model in a finite volume. Sov. Phys. Dokl. 32, 878–879 (1987)
Izergin, A.G., Coker, D.A., Korepin, V.E.: Determinant formula for the six-vertex model. J. Phys. A 25, 4315–4334 (1992)
Jockusch, W., Propp, J., Shor, P.: Random Domino Tilings and the Arctic Circle Theorem. arXiv:math/9801068
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)
Keating, D., Sridhar, A.: Random tilings with the GPU. J. Math. Phys. 59, 094120 (2018)
Keesman, R., Lamers, J.: Numerical study of the \(F\)-model with domain-wall boundaries. Phys. Rev. E 95, 052117 (2017)
Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta Math. 199, 263–302 (2007)
Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006)
Korepin, V.: Calculations of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982)
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)
Kuperberg, G.: Another proof of the alternating-sign matrix conjecture. Int. Math. Res. Not. 139–150, 1996 (1996)
Levin, D.A., Peres, Y., Wilmer, E.L.: With a Chapter by J. G. Propp and D. B. Wilson, Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)
Lieb, E.H.: Residual entropy of square ice. Phys. Rev. Lett. 162, 162–172 (1967)
Lyberg, I., Korepin, V., Ribeiro, G.A.P., Viti, J.: Phase separation in the six-vertex model with a variety of boundary conditions. J. Math. Phys. 59, 053301 (2018)
Mills, W.H., Robbins, D.P., Rumsey, H.: Alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A 34, 340–359 (1983)
Palamarchuk, K., Reshetikhin, N.: The 6-vertex model with fixed boundary conditions. In: Proceedings of Solvay Workshop “Bethe Ansatz: 75 Years Later” (2006)
Petersen, T.K., Speyer, D.: An arctic circle theorem for Groves. J. Combin. Theory Ser. A 111, 137–164 (2005)
Razumov, A.V., Stroganov, Y.G.: Combinatorial nature of the ground-state vector of the \(O(1)\) loop model. Theor. Math. Phys. 138, 333–337 (2004)
Robbins, H.: A remark on Stirling’s formula. Am. Math. Mon. 62, 26–29 (1955)
Sportiello, A.: Simple Approaches to Arctic Curves for Alternating Sign Matrices. Online slides. https://www.ggi.infn.it/talkfiles/slides/talk3694.pdf. Accessed 2 Dec 2019
Reshetikhin, N., Sridhar, A.: Integrability of limit shapes of the six vertex model. Commun. Math. Phys. 356, 535–565 (2017)
Stroganov, Y.G.: Izergin–Korepin determinant at a third root of unity. Theor. Math. Phys. 146, 53–62 (2006)
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)
Sylijuåsen, O.F., Zvonarev, M.B.: Directed-loop Monte Carlo simulations of vertex models. Phys. Rev. E 70, 016118 (2004)
Zeilberger, D.: Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3, R13 (1996)
Zeilberger, D.: Proof of the refined alternating sign matrix conjecture. N. Y. J. Math. 2, 59–68 (1996)
Zinn-Justin, P.: Six-vertex model with domain-wall boundary conditions and one-matrix model. Phys. Rev. E 62, 3411–3418 (2000)
Zinn-Justin, P.: The Influence of Boundary Conditions in the Six-Vertex Model. arXiv:cond-mat/0205192
Acknowledgements
The author heartily thanks Filippo Colomo and Andrea Sportiello for stimulating conversations and enlightening explanations, as well as Alexei Borodin for valuable encouragement and helpful discussions and suggestions. The author would also like to thank two anonymous referees for their helpful comments on an earlier draft of this paper. The author is additionally grateful to the workshop, “Conference on Quantum Integrable Systems, Conformal Field Theories and Stochastic Processes,” held in 2016 at the Institut d’Études Scientifiques de Cargése (funded by NSF Grant DMS:1637087), where he was introduced to the tangent method. This work was partially supported by the NSF Graduate Research Fellowship under Grant Number DGE1144152.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Aggarwal, A. Arctic boundaries of the ice model on three-bundle domains. Invent. math. 220, 611–671 (2020). https://doi.org/10.1007/s00222-019-00938-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-019-00938-6