Discrete Riemann surfaces and the Ising model

Communications in Mathematical Physics

Volumn 218, Issue 1, 2001, Pages 177-216

  Mercat, Christian ^a,b  

^a UNIVERSITÉ DE STRASBOURG   (France)

^b UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035531721     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200000348     Document Type: Article

References (37)

1. [AdABM) Anglès d'Auriac, J.-Ch., Boukraa, S. and Maillard, J.M.: Let's baxterise. J. Stat. Phys. 102, 641-700 (2001). hepth/0003212
2. [BGB] Baake, M, Grimm, U. and Baxter, R.J.: A critical Ising model on the labyrinth. Internat. J. Modern Phys. B 8, 2526 (1994): Perspectives on solvable models. 3579-3600
3. [BGB] Baake, M, Grimm, U. and Baxter, R.J.: A critical Ising model on the labyrinth. Internat. J. Modern Phys. B 8, 2526 (1994): Perspectives on solvable models. 3579-3600
4. [Bax] Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1989; Reprint of the 1982 original
5. [BPZ] Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 2, 333-380 (1984)
6. [BS96] Benjamini, I. and Schramm, O.: Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24, 3, 1219-1238 (1996)
7. [Bus] Buser, P.: Geometry and spectra of compact Riemann surfaces. Boston, MA: Birkhäuser Boston Inc., 1992
8. [CdV90] Colin de Verdière, Y.: Un principe variationnel pour les empilements de cercles. Invent. Math. 104, 655-669 (1991)
9. [CdV96] Colin de Verdière, Y. Gitler, I. and Vertigan, D.: Réseaux électriques planaires, I,I. Comment. Math. Helv. 71, 1, 144-167 (1996)
10. [Cox1] Coxeter, H.S.M.: Introduction to geometry. Wiley Classics Library, New York: John Wiley & Sons Inc., 1989; Reprint of the 1969 edition
11. [DD] Dotsenko, V.S. and Dotsenko, V.S.: Critical behaviour of the phase transition in the 2d Ising model with impurities. Adv. in Phys.32, 2, 129-172 (1983)
12. [Duf] Duffin, R.J.: Basic properties of discrete analytic functions. Duke Math. J. 23, 335-363 (1956)
13. [GS87] Grünbaum, B. and Shephard, G.C.: Tilings and patterns. New York: W. H. Freeman and Company, 1987
14. [Hug] Hughes, B.D.: Random walks and random environments. Vol. 1. New York: The Clarendon Press Oxford University Press, 1995
15. [ID] Itzykson, C. and Drouffe, J.-M.: Statistical field theory, Vol. 2. Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press, 1989
16. [KC] Kadanoff, L.P. and Ceva, H.: Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B (3) 3, 3918-3939 (1971)
17. [K] Kaufman, B: Crystal statistics ii. Partition function evaluated by spinor analysis. Phys. Rev. 76, 1232 (1949)
18. [KO] Kaufman, B and Onsager, L: Crystal statistics iii. Short range order in a binary Ising lattice. Phys. Rev. 76, 1244 (1949)
19. [Ken] Kenyon, R.: Tilings and discrete Dirichlet problems. Israel J. Math. 105, 61-84 (1998)
20. [KW) Kramers, H.A. and Wannier, G.H.: Statistics of the two-dimensional ferromagnet, I. Phys. Rev. (2) 60, 252-262 (1941)
21. [LF] Lelong-Ferrand, J.: Représentation conforme et transformations à intégrale de Dirichlet bornée. Paris: Gauthier-Villars, 1955
22. [McCW] McCoy, B.M. and Wu, T.T.: The two-dimensional Ising model. Cambridge, Massachusetts: Harvard University Press, 1973
23. [McCW81] McCoy, B.M. and Wu, T.T.: Non-linear partial difference equations for the two-spin correlation function of the two-dimensional Ising model. Nucl. Phys. B 180, 89-115 (1981)
24. [M] Mercat, C.: Holomorphie discrète et modèle d'Ising, PhD thesis, Université Louis Pasteur, Strasbourg, France, 1998, under the direction of Daniel Bennequin, Prépublication de l'IRMA, available at http://www-irma.u-strasbg.fr/irma/publications/1998/98014.shtml
25. [Mil] Milnor, J: Spin structures on manifolds. Enseign. Math., II. Ser. 9, 198-203 (1963)
26. [Ons] Onsager, L.: Crystal statistics, I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117-149 (1944)
27. [PS85] Preparata, F.P. and Shamos, M.I.: Computational geometry. Texts and Monographs in Computer Science, New York: Springer-Verlag, 1985
28. [SMJ] Sato, M, Miwa, T and Jimbo, M: Studies on holonomic quantum fields i-iv. Proc. Japan Acad. 53 (A), 1-6 (1977)
29. [Sie] Siegel, C.L.: Topics in complex function theory. Vol. II, Wiley Classics Library, New York: John Wiley & Sons Inc., 1988. Automorphic functions and abelian integrals, translated from the German by A. Shenitzer and M. Tretkoff, with a preface by Wilhelm Magnus, Reprint of the 1971 edition. A Wiley-Interscience Publication
30. [Syo] Syôzi, Itiro: Statistics of kagomé lattice. Progress Theoret. Phys. 6, 306-308 (1951)
31. [Tro] Troyanov, M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2) 32 1, 2, 79-94 (1986)
32. [Uti] Utiyama, T.: Statistics of two-dimensional Ising lattices of chequered types. Progress Theoret. Phys. 6, 907-909 (1951); Letter to the Editor
33. [Veb] Veblen, O.: Analysis situs. 2. edit., New York: American Mathematical Society X, 1931
34. [VoSh] Voevodskiǐ, V.A. and Shabat, G.B.: Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields. Dokl. Akad. Nauk SSSR 304 2, 265-268 (1989)
35. [Wan50) Wannier, G.H.: Antiferromagnetism. The triangular Ising net. Physical Rev. (2) 79, 357-364 (1950)
36. [Whit] Whitney, H.: Product on complexes. Annals of Math. (2) 39, 397-432 (1938)
37. [Yam] Yamamoto, T.: On the crystal statistics of two-dimensional Ising ferromagnets. Progress Theoret. Phys. 6, 533-542 (1951)