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 Probability Surveys > Vol. 8 (2011) open journal systems 

Topics on abelian spin models and related problems

Julien Dubédat, Columbia University

In these notes, we discuss a selection of topics on several models of planar statistical mechanics. We consider the Ising, Potts, and more generally abelian spin models; the discrete Gaussian free field; the random cluster model; and the six-vertex model. Emphasis is put on duality, order, disorder and spinor variables, and on mappings between these models.

AMS 2000 subject classifications: Primary 60G15, 82B20.

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Dubédat, Julien, Topics on abelian spin models and related problems, Probability Surveys, 8, (2011), 374-402 (electronic). DOI: 10.1214/11-PS187.


[1]     Arguin, L.-P. (2002). Homology of Fortuin-Kasteleyn clusters of Potts models on the torus. J. Stat. Phys. 109 301–310. MR1927924

[2]     Baxter, R. J. (1989). Exactly Solved Models in Statistical Mechanics. Academic Press [Harcourt Brace Jovanovich Publishers], London. Reprint of the 1982 original. MR0998375

[3]     Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297–346 (electronic). MR1815214

[4]     Di Francesco, P., Mathieu, P. and Sénéchal, D. (1997). Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York. MR1424041

[5]     Di Francesco, P., Saleur, H. and Zuber, J. B. (1987). Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models. J. Stat. Phys. 49 57–79. MR0923852

[6]     Dubédat, J. Dimers and analytic torsion I. arXiv:1110.2808, 2011.

[7]     Edwards, R. G. and Sokal, A. D. (1988). Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 2009–2012. MR0965465

[8]     Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. I. J. Algebraic Combin. 1 111–132. MR1226347

[9]     Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 219–234. MR1194076

[10]     Fan, C. and Wu, F. Y. (Aug 1970). General lattice model of phase transitions. Phys. Rev. B 2 723–733.

[11]     Fateev, V. A. and Zamolodchikov, A. B. (1982). Self-dual solutions of the star-triangle relations in ZN-models. Phys. Lett. A 92 37–39. MR0677808

[12]     Ferrari, P. L. and Spohn, H. (2006). Domino tilings and the six-vertex model at its free-fermion point. J. Phys. A 39 10297–10306. MR2256593

[13]     Fisher, M. E. and Stephenson, J. (1963). Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers. Phys. Rev. (2) 132 1411–1431. MR0158705

[14]     Gawȩdzki, K. (1999). Lectures on conformal field theory. In Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) 727–805. Amer. Math. Soc., Providence, RI. MR1701610

[15]     Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin. MR2807681

[16]     Glimm, J. and Jaffe, A. (1987). Quantum Physics, 2nd ed. Springer, New York. A functional integral point of view. MR0887102

[17]     Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin. MR1707339

[18]     Grimmett, G. (2006). The Random-cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin. MR2243761

[19]     Grimmett, G. (2010). Probability on Graphs. Institute of Mathematical Statistics Textbooks 1. Cambridge Univ. Press, Cambridge. Random processes on graphs and lattices. MR2723356

[20]     Grünbaum, F. A. (1982). The eigenvectors of the discrete Fourier transform: A version of the Hermite functions. J. Math. Anal. Appl. 88 355–363. MR0667064

[21]     Ikhlef, Y. and Cardy, J. (2009). Discretely holomorphic parafermions and integrable loop models. J. Phys. A 42 102001, 11. MR2485852

[22]     Izergin, A. G., Coker, D. A. and Korepin, V. E. (1992). Determinant formula for the six-vertex model. J. Phys. A 25 4315–4334. MR1181591

[23]     Kadanoff, L. P. and Ceva, H. (1971). Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B (3) 3 3918–3939. MR0389111

[24]     Kasteleyn, P. W. (1961). The statistics of dimers on a lattice. i. the number of dimer arrangements on a quadratic lattice. Physica 27 1209–1225.

[25]     Kenyon, R. Conformal invariance of loops in the double-dimer model. preprint, arXiv:1105.4158, 2011.

[26]     Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128–1137. MR1872739

[27]     Kenyon, R. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Math. Ser. 16 191–230. Amer. Math. Soc., Providence, RI. MR2523460

[28]     Kenyon, R., Okounkov, A. and Sheffield, S. (2006). Dimers and amoebae. Ann. of Math. (2) 163 1019–1056. MR2215138

[29]     Kramers, H. A. and Wannier, G. H. (1941). Statistics of the two-dimensional ferromagnet. I. Phys. Rev. (2) 60 252–262. MR0004803

[30]     Kramers, H. A. and Wannier, G. H. (1941). Statistics of the two-dimensional ferromagnet. II. Phys. Rev. (2) 60 263–276. MR0004804

[31]     Lieb, E. H. (Oct 1967). Residual entropy of square ice. Phys. Rev. 162 162–172.

[32]     McCoy, B. and Wu, T. The two-dimensional Ising model. Harvard Univ. Press, Boston, MA, 1973.

[33]     Mercat, C. (2001). Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218 177–216. MR1824204

[34]     Nienhuis, B. (1984). Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34 731–761. MR0751711

[35]     Nienhuis, B. and Knops, H. J. F. (1872–1875, Aug). Spinor exponents for the two-dimensional potts model. Phys. Rev. B 32 1985.

[36]     Palmer, J. (2007). Planar Ising Correlations. Progress in Mathematical Physics 49. Birkhäuser, Boston, MA. MR2332010

[37]     Pinson, H. T. (1994). Critical percolation on the torus. J. Stat. Phys. 75 1167–1177. MR1285297

[38]     Reshetikhin, N. (2010). Lectures on the integrability of the six-vertex model. In Exact Methods in Low-dimensional Statistical Physics and Quantum Computing 197–266. Oxford Univ. Press, Oxford. MR2668647

[39]     Riva, V. and Cardy, J. (2006). Holomorphic parafermions in the Potts model and stochastic Loewner evolution. J. Stat. Mech. Theory Exp. 12 P12001, 19 pp. (electronic). MR2280251

[40]     Rudin, W. (1990). Fourier Analysis on Groups. Wiley Classics Library. Wiley, New York. Reprint of the 1962 original, A Wiley-Interscience Publication. MR1038803

[41]     Savit, R. (1982). Duality transformations for general abelian systems. Nuclear Phys. B 200 233–248. MR0643588

[42]     Simon, B. (1974). The P(ϕ)2 Euclidean (quantum) Field Theory. Princeton Univ. Press, Princeton, N.J. Princeton Series in Physics. MR0489552

[43]     Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II 1421–1451. Eur. Math. Soc., Zürich. MR2275653

[44]     Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435–1467. MR2680496

[45]     Thurston, W. P. (1990). Conway’s tiling groups. Amer. Math. Monthly 97 757–773. MR1072815

[46]     van Beijeren, H. (May 1977). Exactly solvable model for the roughening transition of a crystal surface. Phys. Rev. Lett. 38 993–996.

[47]     Werner, W. (2004). Random planar curves and Schramm-Loewner evolutions. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1840 107–195. Springer, Berlin. MR2079672

[48]     Wu, F. Y. and Wang, Y. K. (1976). Duality transformation in a many-component spin model. J. Math. Phys. 17 439–440.

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