Probability Surveys

Topics on abelian spin models and related problems

Julien Dubédat

Full-text: Open access

Abstract

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.

Article information

Source
Probab. Surveys, Volume 8 (2011), 374-402.

Dates
First available in Project Euclid: 30 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.ps/1325264816

Digital Object Identifier
doi:10.1214/11-PS187

Mathematical Reviews number (MathSciNet)
MR2861134

Zentralblatt MATH identifier
1243.82020

Subjects
Primary: 60G15: Gaussian processes 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Citation

Dubédat, Julien. Topics on abelian spin models and related problems. Probab. Surveys 8 (2011), 374--402. doi:10.1214/11-PS187. https://projecteuclid.org/euclid.ps/1325264816


Export citation

References

  • [1] Arguin, L.-P. (2002). Homology of Fortuin-Kasteleyn clusters of Potts models on the torus. J. Stat. Phys. 109 301–310.
  • [2] Baxter, R. J. (1989). Exactly Solved Models in Statistical Mechanics. Academic Press [Harcourt Brace Jovanovich Publishers], London. Reprint of the 1982 original.
  • [3] Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297–346 (electronic).
  • [4] Di Francesco, P., Mathieu, P. and Sénéchal, D. (1997). Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York.
  • [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.
  • [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.
  • [8] Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. I. J. Algebraic Combin. 1 111–132.
  • [9] Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 219–234.
  • [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.
  • [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.
  • [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.
  • [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.
  • [15] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd ed. de Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
  • [16] Glimm, J. and Jaffe, A. (1987). Quantum Physics, 2nd ed. Springer, New York. A functional integral point of view.
  • [17] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [18] Grimmett, G. (2006). The Random-cluster Model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 333. Springer, Berlin.
  • [19] Grimmett, G. (2010). Probability on Graphs. Institute of Mathematical Statistics Textbooks 1. Cambridge Univ. Press, Cambridge. Random processes on graphs and lattices.
  • [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.
  • [21] Ikhlef, Y. and Cardy, J. (2009). Discretely holomorphic parafermions and integrable loop models. J. Phys. A 42 102001, 11.
  • [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.
  • [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.
  • [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.
  • [27] Kenyon, R. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Math. Ser. 16 191–230. Amer. Math. Soc., Providence, RI.
  • [28] Kenyon, R., Okounkov, A. and Sheffield, S. (2006). Dimers and amoebae. Ann. of Math. (2) 163 1019–1056.
  • [29] Kramers, H. A. and Wannier, G. H. (1941). Statistics of the two-dimensional ferromagnet. I. Phys. Rev. (2) 60 252–262.
  • [30] Kramers, H. A. and Wannier, G. H. (1941). Statistics of the two-dimensional ferromagnet. II. Phys. Rev. (2) 60 263–276.
  • [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.
  • [34] Nienhuis, B. (1984). Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34 731–761.
  • [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.
  • [37] Pinson, H. T. (1994). Critical percolation on the torus. J. Stat. Phys. 75 1167–1177.
  • [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.
  • [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).
  • [40] Rudin, W. (1990). Fourier Analysis on Groups. Wiley Classics Library. Wiley, New York. Reprint of the 1962 original, A Wiley-Interscience Publication.
  • [41] Savit, R. (1982). Duality transformations for general abelian systems. Nuclear Phys. B 200 233–248.
  • [42] Simon, B. (1974). The P(ϕ)2 Euclidean (quantum) Field Theory. Princeton Univ. Press, Princeton, N.J. Princeton Series in Physics.
  • [43] Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II 1421–1451. Eur. Math. Soc., Zürich.
  • [44] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 1435–1467.
  • [45] Thurston, W. P. (1990). Conway’s tiling groups. Amer. Math. Monthly 97 757–773.
  • [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.
  • [48] Wu, F. Y. and Wang, Y. K. (1976). Duality transformation in a many-component spin model. J. Math. Phys. 17 439–440.