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These lecture notes provide a (almost) self-contained account on conformal invariance of the planar critical Ising and FK-Ising models.
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1920
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H.B. Phillips and N. Wiener, Nets and the Dirichlet problem. , Math. J. of Math. 2
1923
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E. Ising, Beitrag zur theorie des ferromagnetismus. , Z. Phys. 31
1925
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G. Bouligand, Sur le problème de Dirichlet. , Ann. Soc. Pol. Math. 4
1926
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L. Lusternik, über einege Anwendungen der direkten Methoden in Variationsrechnung. , Recueil de la Société Mathématique de Moscou, 1926, pp. 173–201
1926
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1928
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W. Heisenberg, Zür theorie des ferromagnetismus , Zeitschrift für Physik A Hadrons and Nuclei 49
1928
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R. Peierls, On Ising’s model of ferromagnetism. , Math. Proc. Camb. Phil. Soc. 32
1936
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1941
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1944
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B. Kaufman and L. Onsager, Crystal statistics. IV. long-range order in a binary crystal , 1950
1950
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M. Kac and J. C. Ward, A combinatorial solution of the two-dimensional ising model , Phys. Rev 88
1952
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C.N. Yang, The spontaneous magnetization of a two-dimensional Ising model , Phys. Rev. (2) 85
1952
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J. Lelong-Ferrand, Représentation conforme et transformations à intégrale de Dirichlet bornée , Gauthier-Villars, Paris, 1955
1955
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R. J. Duffin, Basic properties of discrete analytic functions , Duke Math. J. 23
1956
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S. Sherman, Combinatorial aspects of the Ising model for ferromagnetism. I. A conjecture of Feynman on paths and graphs , J. Mathematical Phys. 1
1960
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P. W. Kasteleyn, The statistics of dimers on a lattice. , PhysicA 27
1961
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N. V. Vdovichenko, A calculation of the partition function for a plane dipole lattice , Soviet Physics JETP 20
1965
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M. Fisher, On the dimer solution of planar Ising models , Journal of Mathematical Physics 7
1966
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L. P. Kadanoff, Scaling laws for Ising model near t c t_{c} , Physics 2
1966
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A.Z. Patashinskii and V.L. Pokrovskii, Behavior of ordered systems near the transition point , Soviet Physics JETP 23
1966
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E.H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice , Physical Review Letters 18
1967
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C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets , Comm. Math. Phys. 22
1971
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V.L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. ii. quantum systems , Soviet Journal of Experimental and Theoretical Physics 34
1972
Cited alongside, same era.
R. L. Dobrushin, Gibbs state, describing the coexistence of phases in the three-dimensional ising model. , PTRF 17
1972
Cited alongside, same era.
C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models , Physica 57
1972
Cited alongside, same era.
J.M. Kosterlitz and D.J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems , Journal of Physics C: Solid State Physics 6
1973
Cited alongside, same era.
B.M. McCoy and T.T. Wu, The two-dimensional Ising model , Harvard University Press, Cambridge, MA, 1973
1973
G. F. Lawler, O. Schramm, and W. Werner, On the scaling limit of planar self-avoiding walk , Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 339–364
2004
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A.I. Bobenko, C. Mercat, and Y. B. Suris, Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function , J. Reine Angew. Math. 583
2005
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R. Kenyon and J.-M. Schlenker, Rhombic embeddings of planar quad-graphs , Trans. Amer. Math. Soc. 357
2005
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O. Schramm and S. Sheffield, Harmonic explorer and its convergence to SLE 4 {\rm SLE}_{4} , Ann. Probab. 33
2005
Later among the works it cites.
K. Stephenson, Introduction to circle packing , Cambridge University Press, Cambridge, 2005, The theory of discrete analytic functions
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Cited alongside, same era.
M. Aizenman, Translation invariance and instability of phase coexistence in the two-dimensional Ising system , Comm. Math. Phys. 73
1980
Cited alongside, same era.
D. Domany, D. Mukamel, B. Nienhuis, and A. Schwimmer, Duality relations and equivalences for models with O(N) and cubic symmetry , Nuclear Physics B 190
1981
Cited alongside, same era.
Y. Higuchi, On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model , Random fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam, 1981, pp. 517–534
1981
Cited alongside, same era.
R.J. Baxter, Exactly solved models in statistical mechanics , Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989, Reprint of the 1982 original
1982
Cited alongside, same era.
B. Nienhuis, Exact critical point and critical exponents of o ( n ) o(n) models in two dimensions. , Phys. Rev. Lett. 49
1982
Cited alongside, same era.
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory , Nuclear Phys. B 241
1984
Cited alongside, same era.
W. P. Thurston, Zippers and univalent functions , The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., vol. 21, Amer. Math. Soc., Providence, RI, 1986, pp. 185–197
1986
Cited alongside, same era.
2005
Later among the works it cites.
G.R. Grimmett, The random-cluster model , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333, Springer-Verlag, Berlin, 2006
2006
Later among the works it cites.
R.J. Messikh, Surface tension near criticality of the 2d-Ising model. , arXiv:math/0610636 (2006), 24 pages
2006
Later among the works it cites.
V. Riva and J.L. Cardy, Holomorphic parafermions in the Potts model and stochastic Loewner evolution , J. Stat. Mech. Theory Exp. (2006), no. 12, P12001, 19 pp. (electronic)
2006
Later among the works it cites.
J. Palmer, Planar Ising correlations , Progress in Mathematical Physics, vol. 49, Birkhäuser Boston Inc., Boston, MA, 2007
2007
Later among the works it cites.
M. A. Rajabpour and J.L. Cardy, Discretely holomorphic parafermions in lattice Z N Z_{N} models , J. Phys. A 40
2007
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A. I. Bobenko and Y. B. Suris, Discrete differential geometry , Graduate Studies in Mathematics, vol. 98, American Mathematical Society, Providence, RI, 2008, Integrable structure
2008
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D. Chelkak and S. Smirnov, Discrete complex analysis on isoradial graphs , to appear in Adv. in Math. (2008), 35 pages
2008
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Y. Ikhlef and J.L. Cardy, Discretely holomorphic parafermions and integrable loop models , J. Phys. A 42
2009
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W. Werner, Percolation et modèle d’Ising , Cours Spécialisés [Specialized Courses], vol. 16, Société Mathématique de France, Paris, 2009
2009
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V. Beffara and H. Duminil-Copin, The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 q\geq 1 , to appear in PTRF (2010), 25 pages
2010
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C. Boutillier and B. de Tilière, The critical Z Z -invariant Ising model via dimers: the periodic case , PTRF 147
2010
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D. Cimasoni, A generalized Kac-Ward formula , to appear J. Stat. Mech. Theory Exp. (2010), 23 pages
2010
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L. Coquille and Y. Velenik, A finite-volume version of Aizenman-Higuchi theorem for the 2D Ising model , to appear in PTRF (2010), 16 pages
2010
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H. Duminil-Copin, C. Hongler, and P. Nolin, Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model , to appear in Communications in Pure and Applied Mathematics (2010), 33 pages
2010
Later among the works it cites.
A. Kemppainen and S. Smirnov, Random curves, scaling limits and loewner evolutions , in preparation (2010)
2010
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E. Lubetzky and A. Sly, Critical Ising on the square lattice mixes in polynomial time , 2010, 26 pages
2010
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2010
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H. Duminil-Copin and S. Smirnov, The connective constant of the honeycomb lattice equals 2 + 2 \sqrt{2+\sqrt{2}} , to appear in Ann. of Math. (2011), 12 pages
2011
Closest in time.
C. Hongler and K. Kytolä, Dipolar SLE in the Ising model with plus/minus/free boundary conditions , 2011, 82 pages
2011
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R. G. Edwards and A. D. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm , Phys. Rev. D (3) 38
2012
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C. Hongler and S. Smirnov, The energy density in the planar ising model , to appear in Acta Math. (2012)
2012
Closest in time.