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Probability in Complex Physical Systems pp 491–512Cite as

Statistical Mechanics on Isoradial Graphs

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Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 11))

Abstract

Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.

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Notes

  1. 1.

    One can define in a similar way a Laplacian on G  ∗ by restricting the same operator \(\partial \bar{\partial }\)to G  ∗ .

  2. 2.

    The correspondence can be extended to surfaces with boundary by including in addition to the closed contours a certain number of paths connected to boundary.

  3. 3.

    On the boundary, we put only half-rhombi such that only “black” vertices are exposed on the boundary.

  4. 4.

    Also known as the Fortuin–Kasteleyn percolation.

References

  1. Au-Yang, H., Perk, J.H.H.: Critical correlations in a Z-invariant inhomogeneous Ising model. Phys. A 144(1), 44–104 (1987)

    Article  MathSciNet  Google Scholar 

  2. Au-Yang, H., Perk, J.H.H.: Q-dependent susceptibilities in ferromagnetic quasiperiodic Z-invariant Ising models. J. Stat. Phys. 127, 265–286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Au-Yang, H., Perk, J.H.H.: Q-dependent susceptibility in Z-invariant pentagrid Ising model. J. Stat. Phys. 127, 221–264 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauer, M., Bernard, D.: CFTs of SLEs: The radial case. Phys. Lett. B 583(3–4), 324–330 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Baxter, R.J.: Eight-vertex model in lattice statistics. Phys. Rev. Lett. 26(14), 832–833 (1971)

    Article  Google Scholar 

  6. Baxter, R.J.: Free-fermion, checkerboard and Z-invariant lattice models in statistical mechanics. Proc. Roy. Soc. Lond. Ser. A 404(1826), 1–33 (1986)

    Article  MathSciNet  Google Scholar 

  7. Baxter, R.J.: Perimeter Bethe Ansatz. J. Phys. A 20(9), 2557 (1987)

    Article  MathSciNet  Google Scholar 

  8. Baxter, R.J.: Exactly solved models in statistical mechanics. Academic, London [Harcourt Brace Jovanovich Publishers] (1989) Reprint of the 1982 original

    Google Scholar 

  9. Beffara, V.: Is critical 2D percolation universal? In: In and out of equilibrium. 2, vol. 60 of Progr. Probab., pp. 31–58. Birkhäuser, Basel (2008)

    Google Scholar 

  10. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241(2), 333–380 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bobenko, A.I., Suris, Y.B.: Discrete differential geometry, vol. 98 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2008); Integrable structure

    Google Scholar 

  12. Bobenko, A.I., Mercat, C., Suris, Y.B.: Linear and nonlinear theories of discrete analytic functions.. Integrable structure and isomonodromic Green’s function. J. Reine Angew. Math. 583, 117–161 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boutillier, C., de Tilière, B.: The critical Z-invariant Ising model via dimers: The periodic case. Probab. Theory Relat. Fields 147, 379–413 (2010) 10.1007/s00440-009-0210-1

    Article  MATH  Google Scholar 

  14. Boutillier, C., de Tilière, B.: The critical Z-invariant Ising model via dimers: locality properties. Comm. Math. Phys. 301, 473–516 (2011) 10.1007/s00220-010-1151-3

    Article  MathSciNet  MATH  Google Scholar 

  15. Bücking, U.: Approximation of conformal mappings by circle patterns. Geom. Dedicata 137, 163–197 (2008) 10.1007/s10711-008-9292-7

    Article  MathSciNet  MATH  Google Scholar 

  16. Burton, R., Pemantle, R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21(3), 1329–1371 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. in Math. 2281590–1630 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Inv. Math.

    Google Scholar 

  19. Cimasoni, D.: A generalized Kac-Ward formula. J. Stat. Mech. Theory Exp. 2010(07), P07023 (2010)

    Article  Google Scholar 

  20. Cimasoni, D.: Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices. J. Eur. Math. Soc. to appear

    Google Scholar 

  21. de Tilière, B.: Partition function of periodic isoradial dimer models. Probab. Theory Relat. Fields 138(3–4), 451–462 (2007)

    Article  MATH  Google Scholar 

  22. de Tilière, B.: Quadri-tilings of the plane. Probab. Theory Relat. Fields 137(3–4), 487–518 (2007)

    MATH  Google Scholar 

  23. de Tilière, B.: Scaling limit of isoradial dimer models and the case of triangular quadri-tilings. Ann. I. H. Poincare B 43(6), 729–750 (2007)

    Article  MATH  Google Scholar 

  24. Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal field theory. Graduate Texts in Contemporary Physics. Springer, New York (1997)

    Book  MATH  Google Scholar 

  25. Duffin, R.J.: Potential theory on a rhombic lattice. J. Combin. Theory 5, 258–272 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fisher, M.E.: On the dimer solution of planar Ising models. J. Math. Phys. 7, 1776–1781 (1966)

    Article  Google Scholar 

  27. Friedrich, R., Werner, W.: Conformal restriction, highest-weight representations and SLE. Comm. Math. Phys. 243(1), 105–122 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Grimmett, G.: The random-cluster model, vol. 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)

    Google Scholar 

  29. Ishikawa, M., Wakayama, M.: Minor summation formulas of Pfaffians, survey and a new identity. In: Combinatorial methods in representation theory (Kyoto, 1998), vol. 28 of Adv. Stud. Pure Math., pp. 133–142. Kinokuniya, Tokyo (2000)

    Google Scholar 

  30. Kasteleyn, P.W.: The statistics of dimers on a lattice: I. the number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)

    Google Scholar 

  31. Kasteleyn, P.W.: Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, pp. 43–110. Academic, London (1967)

    Google Scholar 

  32. Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33(5), 591–618 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kenyon, R.: An introduction to the dimer model. Lecture notes ICTP (2002) ICTP Lecture note series volume XVII (ISBN 92-95003-25-X) May 2004

    Google Scholar 

  35. Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kirchhoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. 148, 497–508 (1847)

    Article  Google Scholar 

  37. Kirchhoff, G.: An exploration of the permanent-determinant method. Electron. J. Combin. 5(46), 34 (1998)

    MathSciNet  Google Scholar 

  38. Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60(3), 252–262 (1941)

    Article  MathSciNet  Google Scholar 

  39. Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60(3), 263–276 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)

    Google Scholar 

  41. Lebowitz, J.L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Comm. Math. Phys. 25, 276–282 (1972)

    Article  MathSciNet  Google Scholar 

  42. Lenz, W.: Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern. Physik. Z. 21, 613–615 (1920)

    Google Scholar 

  43. Lieb, E.H.: Exact solution of the problem of the entropy of two-dimensional ice. Phys. Rev. Lett. 18, 692–694 (1967)

    Article  Google Scholar 

  44. Martìnez, J.R.R.: Correlation functions for the Z-invariant Ising model. Phys. Lett. A 227(3–4), 203–208 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. Martìnez, J.R.R.: Multi-spin correlation functions for the Z-invariant Ising model. Phys. A 256(3–4), 463–484 (1998)

    Google Scholar 

  46. McCoy, B., Wu, T.T.: The Two-Dimensional Ising model. Harvard University Press, Cambridge (1973)

    MATH  Google Scholar 

  47. Mercat, C.: Discrete Riemann surfaces and the Ising model. Comm. Math. Phys. 218(1), 177–216 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Perk, J.H.H., Au-Yang, H.: Yang-Baxter equations. In: Françoise, J.P., Naber, G.L., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics, pp. 465–473. Academic, Oxford (2006)

    Chapter  Google Scholar 

  49. Smirnov, S.: Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)

    Article  MATH  Google Scholar 

  50. Smirnov, S.: Conformal invariance in random cluster models. I. holomorphic fermions in the Ising model. Ann. Math. 172(2), 1435–1467 (2010)

    MATH  Google Scholar 

  51. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics-an exact result. Philos. Mag. 6(68), 1061–1063 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  52. Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 place Jussieu, 75005, Paris, France

    Cédric Boutillier & Béatrice de Tilière

Authors
  1. Cédric Boutillier
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  2. Béatrice de Tilière
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Corresponding author

Correspondence to Cédric Boutillier .

Editor information

Editors and Affiliations

  1. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Jean-Dominique Deuschel

  2. Faculty of Mathematics, University of Bielefeld, P.O. Box 10 01 31, Bielefeld, 33501, Germany

    Barbara Gentz

  3. for Applied Analysis and Stochastics, Weierstrass Institute Berlin, Mohrenstr. 39, Berlin, 10117, Germany

    Wolfgang König

  4. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Max von Renesse

  5. Institute for Mathematics, TU Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany

    Michael Scheutzow

  6. Institute for Mathematical, Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8-10/105-1, Vienna, 1040, Austria

    Uwe Schmock

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Boutillier, C., de Tilière, B. (2012). Statistical Mechanics on Isoradial Graphs. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_20

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