Numerical stu\-dy of the 6-vertex model with domain wall boundary conditions
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, p. 1847-1869
A Markov process converging to a random state of the 6-vertex model is constructed. It is used to show that a droplet of c-vertices is created in the antiferromagnetic phase and that the shape of this droplet has four cusps.
Nous construisons un processus de Markov qui converge vers un état aléatoire du modèle 6- vertex. Ensuite, nous l’utilisons pour faire apparaître la création dans la phase antiferromagnétique d’une goutelette constituée de sommets de type c et dont la forme possède 4 pointes.
DOI : https://doi.org/10.5802/aif.2144
Classification:  82-08,  82B20,  82B23
Keywords: 6-vertex, Markov chain, random sampling, Monte Carlo
@article{AIF_2005__55_6_1847_0,
     author = {Allison, David and Reshetikhin, Nicolai},
     title = {Numerical stu\-dy of the 6-vertex model with domain wall boundary conditions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     pages = {1847-1869},
     doi = {10.5802/aif.2144},
     zbl = {02230060},
     mrnumber = {2187938},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_6_1847_0}
}
Allison, David; Reshetikhin, Nicolai. Numerical stu\-dy of the 6-vertex model with domain wall boundary conditions. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 1847-1869. doi : 10.5802/aif.2144. http://www.numdam.org/item/AIF_2005__55_6_1847_0/

[1] H. Cohn; M. Larsen; J. Propp The shape of a typical boxed plane partition, New York J. Math., Tome 4 (1998), pp. 137-165 | MR 1641839 | Zbl 0908.60083

[2] H. Cohn; N. Elkis; J. Propp Local statistics of random domino tilings of the Aztec diamons, Duke Math. J., Tome 85 (1996), pp. 117-166 | MR 1412441 | Zbl 0866.60018

[3] R.J. Baxter Exactly Solved Models in Statistical Mechanics, Academic Press, San Diego (1982) | MR 690578 | Zbl 0538.60093

[4] K. Eloranta Diamond Ice, J. Statist. Phys., Tome 96 (1999) no. 5-6, pp. 1091-1109 | MR 1722988 | Zbl 01404169

[5] A. Izergin Partition function of the 6-vertex model in a finite volume, Sov. Phys. Dokl., Tome 32 (1987), p. 878-879 | Zbl 0875.82015

[6] R. Kenyon An introduction to the dimer models (math.CO/0310326, http://arxiv.org/abs/math.CO/0310326) | Zbl 1076.82025

[7] V. Korepin; P. Zinn-Justin Thermodynamic limit of the six-vertex model withdomain wall boundary conditions (cond–mat/0004250, http://arxiv.org/abs/cond-mat/0004250) | Zbl 0956.82008

[8] V.E. Korepin Calculation of norms of Bethe wave functions, Comm. Math. Phys., Tome 86 (1982) no. 3, pp. 391-418 | Article | MR 677006 | Zbl 0531.60096

[9] G. Kuperberg Another proof of the alternating-sign matrix conjecture, Int. Math. Res. Notes, Tome 3 (1996), pp. 139-150 | MR 1383754 | Zbl 0859.05027

[10] E. Lieb; F. Wu; C. Domb And M.S. Green Two dimensional ferroelectric models, Phase transitions and critical phenomena, Academic Press (1972)

[11] R. Kenyon; A. Okounkov; S. Sheffield Dimers and amoebae (math–ph/0311005, http://arxiv.org/abs/math-ph/0311005) | Zbl 05051319

[12] A. Okounkov; N. Reshetikhin Random skew plane partitions and the Pearcey process (math.CO/0503508, http://arxiv.org/abs/math.CO/0503508)

[13] A. Okounkov; N. Reshetikhin Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc., Tome 16 (2003) no. 3, pp. 581-603 | Article | MR 1969205 | Zbl 1009.05134

[14] J. Propp; D. Wilson Coupling from the past: a user's guide, Microsurveys in Discrete Probability (Princeton, NJ, 1997), AMS (DIMACS Ser. Discrete Math. Theoret. Comp. Sci.) Tome 41 (1998), pp. 181-192 | Zbl 0916.65147

[15] A. Sinclair Algorithms for Random Generation and Counting, Birkhauser, Boston (1993) | MR 1201590 | Zbl 0780.68096

[16] O.F. Syljuasen; M.B. Zvonarev Directed-loop Monte Carlo simulations of vertex models (cond-mat/0401491)

[17] A.V. Razumov; Yu. Stroganov Combinatorial structure of the ground state of O(1) loop model (math.CO/0104216, http://arxiv.org/abs/math.CO/0104216)

[18] Asymptotic combinatorics withapplications to mathematical physics, Springer, (ed. by A.M. Vershik), Tome 1815 (2003) | Zbl 1014.00010

[19] P. Zinn-Justin Six-vertex model with domain wall boundary conditions and one-matrix model, Phys. Rev. E, Tome 62 (2000) no. 3, part A, pp. 3411-3418 | Article | MR 1788950

[20] J.-B. Zuber On the counting of fully packed loop configurations. Some new conjectures (math-ph/0309057, http://arxiv.org/abs/math-ph/0309057) | Zbl 1054.05011