Emergent planarity in two-dimensional Ising models with finite-range Interactions

  • Michael AizenmanEmail author
  • Hugo Duminil-Copin
  • Vincent Tassion
  • Simone Warzel


The known Pfaffian structure of the boundary spin correlations, and more generally order–disorder correlation functions, is given a new explanation through simple topological considerations within the model’s random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on \({\mathbb {Z}}^2\) with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.



This work was supported in parts by the NSF Grants PHY-1305472, DMS-1613296, the NCCR SwissMAP, the IDEX grant from Paris-Saclay, the ERC CriBLaM, and a Princeton University/University of Geneva partnership fund.

Supplementary material


  1. 1.
    Aizenman, M.: Geometric analysis of \(\varphi ^{4}\) fields and Ising models. Commun. Math. Phys. 86(1), 1–48 (1982)zbMATHGoogle Scholar
  2. 2.
    Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47, 343–374 (1987)Google Scholar
  3. 3.
    Aizenman, M., Burchard, A.: Hölder regularity and dimension bounds for random curves. Duke Math. J. 99(3), 419–453 (1999)zbMATHGoogle Scholar
  4. 4.
    Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in one-dimensional \(1/|x-y|^2\) Ising and Potts models. J. Stat. Phys. 50, 1–40 (1988)zbMATHGoogle Scholar
  5. 5.
    Aizenman, M., Duminil-Copin, H., Sidoravicius, V.: Random currents and continuity of Ising model’s spontaneous magnetization. Commun. Math. Phys. 334, 719–742 (2015)zbMATHGoogle Scholar
  6. 6.
    Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44, 393–454 (1986)zbMATHGoogle Scholar
  7. 7.
    Aizenman, A., Fernández, Roberto: Critical exponents for long-range interactions. Lett. Math. Phys. 16, 39–49 (1988)zbMATHGoogle Scholar
  8. 8.
    Aizenman, M., Warzel, S.: Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function. J. Stat. Phys. 173, 1755–1778 (2018)zbMATHGoogle Scholar
  9. 9.
    Baxter, R.J.: Solvable eight-vertex model on an arbitrary planar lattice. Philos. Trans. R. Soc. Lond. Ser. A 289(1359), 315–346 (1978)Google Scholar
  10. 10.
    Benoist, S., Hongler, C.: The scaling limit of critical Ising interfaces is CLE(3) (2016). arXiv:1604.06975
  11. 11.
    Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys. 121, 501–505 (1989)zbMATHGoogle Scholar
  12. 12.
    Chelkak, D., Cimasoni, D., Kassel, A.: Revisiting the combinatorics of the 2D Ising model. Ann. Inst. Henri Poincaré D (2016) (to appear) Google Scholar
  13. 13.
    Chelkak, D., Duminil-Copin, H., Hongler, C.: Crossing probabilities in topological rectangles for the critical planar FK-Ising model. Electron. J. Probab. 5, 28 (2016)zbMATHGoogle Scholar
  14. 14.
    Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A., Smirnov, S.: Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Acad. Sci. Paris Math. 352(2), 157–161 (2014)zbMATHGoogle Scholar
  15. 15.
    Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189, 515–580 (2012)zbMATHGoogle Scholar
  16. 16.
    Duminil-Copin, H.: Lectures on the Ising and Potts models on the hypercubic lattice (2017). arXiv:1707.00520
  17. 17.
    Duminil-Copin, H., Hongler, C., Nolin, P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Commun. Pure Appl. Math. 64(9), 1165–1198 (2011)zbMATHGoogle Scholar
  18. 18.
    Duminil-Copin, H., Lis, M.: On the double random current nesting field (2017). arXiv:1712.02305
  19. 19.
    Duminil-Copin, H., Sidoravicius, V., Tassion, V.: Absence of infinite cluster for critical Bernoulli percolation on slabs. Commun. Pure Appl. Math. 69(7), 1397–1411 (2016)zbMATHGoogle Scholar
  20. 20.
    Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343, 725–745 (2016)zbMATHGoogle Scholar
  21. 21.
    Duminil-Copin, H., Tassion, V.: RSW and box-crossing property for planar percolation. In: Proceedings of the International Congress of Mathematical Physics (2016)Google Scholar
  22. 22.
    Fisher, M.E.: On the dimer solution of planar Ising models. J. Math. Phys. 7(10), 1776–1781 (1966)Google Scholar
  23. 23.
    Georgii, H.-O.: Gibbs measures and phase transitions, 2nd edn. volume 9 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (2011)Google Scholar
  24. 24.
    Giuliani, A., Greenblatt, R., Mastropietro, V.: The scaling limit of the energy correlations in non-integrable Ising models. J. Math. Phys. 53(9), 095214 (2012)zbMATHGoogle Scholar
  25. 25.
    Griffiths, R.: Correlation in Ising ferromagnets I, II. J. Math. Phys. 8, 478–489 (1967)Google Scholar
  26. 26.
    Groeneveld, J., Boel, R.J., Kasteleyn, P.W.: Corrrelation-function identities for general planar Ising systems. Physica 93A, 138–154 (1978)Google Scholar
  27. 27.
    Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11, 790–795 (1970)Google Scholar
  28. 28.
    Grimmett, G.: The Random-Cluster Model, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 333. Springer, Berlin (2006)Google Scholar
  29. 29.
    Grimmett, G., Janson, S.: Random even subgraphs. Electron. J. Probab. 16(1), 1–19 (2009)zbMATHGoogle Scholar
  30. 30.
    Hegerfeldt, G.C.: Correlation inequalities for Ising ferromagnets with symmetries. Commun. Math. Phys. 57(3), 259–266 (1977)Google Scholar
  31. 31.
    Hurst, C.A., Green, H.S.: New solution of the Ising problem for a rectangular lattice. J. Chem. Phys. 33, 1059–1062 (1960)Google Scholar
  32. 32.
    Kadanoff, L.P.: Spin–spin correlation in the two-dimensional Ising model. Nuovo Cimento 44, 276–305 (1966)Google Scholar
  33. 33.
    Kadanoff, L.P., Ceva, H.: Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B 3, 3918–3939 (1971)Google Scholar
  34. 34.
    Kasteleyn, P.W.: Dimer statistics and phase transitions. J. Math. Phys. 4, 287–293 (1963)Google Scholar
  35. 35.
    Kaufman, B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev. 76, 1232–1243 (1949)zbMATHGoogle Scholar
  36. 36.
    Lieb, E.H.: A refinement of Simon’s correlation inequality. Commun. Math. Phys. 77, 127–135 (1980)Google Scholar
  37. 37.
    Lis, M.: The planar Ising model and total positivity. J. Stat. Phys. 166(1), 72–89 (2017)zbMATHGoogle Scholar
  38. 38.
    Lupu, T., Werner, W.: A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field. Electron. Commun. Probab. 21, paper no. 13, 7 (2016)Google Scholar
  39. 39.
    Manolescu, I., Raoufi, A.: The phase transitions of the random-cluster and Potts models on slabs with \( q\ge 1\) are sharp (2016). arXiv:1604.01299
  40. 40.
    McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)zbMATHGoogle Scholar
  41. 41.
    McCoy, B.M., Perk, J.H.H., Wu, T.T.: Ising field theory: quadratic difference equations for the \(n\)-point Green’s functions on the square lattice. Phys. Rev. Lett. 46, 757–760 (1981)Google Scholar
  42. 42.
    Messager, A., Miracle-Sol, S.: Correlation functions and boundary conditions in the Ising ferromagnet. J. Stat. Phys. 17(4), 245–262 (1977)Google Scholar
  43. 43.
    Newman, C.M., Tassion, V., Wu, W.: Critical percolation and the minimal spanning tree in slabs. Commun. Pure Appl. Math. 70(11), 2084–2120 (2017)zbMATHGoogle Scholar
  44. 44.
    Pinson, H., Spencer, T.: Universality and the two-dimensional Ising model (unpublished preprint) Google Scholar
  45. 45.
    Onsager, L.: Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. (2) 65, 117–149 (1944)zbMATHGoogle Scholar
  46. 46.
    Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43, 39–48 (1978)zbMATHGoogle Scholar
  47. 47.
    Schrader, R.: New correlation inequalities for the Ising model and \(P(\phi )\) theories. Phys. Rev. B 15, 2798 (1977)Google Scholar
  48. 48.
    Schultz, T.D., Mattis, D., Lieb, E.H.: Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36, 856 (1964)Google Scholar
  49. 49.
    Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227–245 (1978)zbMATHGoogle Scholar
  50. 50.
    Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys. 77, 111–126 (1980)Google Scholar
  51. 51.
    Tassion, V.: Crossing probabilities for Voronoi percolation. Ann. Probab. 44(5), 3385–3398 (2016)zbMATHGoogle Scholar
  52. 52.
    van der Waerden, B.L.: Die lange Reichweite der regelmassigen Atomanordnung in Mischkristallen. Z. Physik 118, 473–488 (1941)zbMATHGoogle Scholar
  53. 53.
    Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 2(85), 808–816 (1952)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Michael Aizenman
    • 1
    Email author
  • Hugo Duminil-Copin
    • 2
    • 3
  • Vincent Tassion
    • 4
  • Simone Warzel
    • 5
  1. 1.Princeton UniversityPrincetonUSA
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance
  3. 3.Université de GenèveGenevaSwitzerland
  4. 4.ETH ZürichZurichSwitzerland
  5. 5.Technische Universität MünchenGarchingGermany

Personalised recommendations