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

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## Abstract

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.

## Notes

### Acknowledgements

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

## References

- 1.Aizenman, M.: Geometric analysis of \(\varphi ^{4}\) fields and Ising models. Commun. Math. Phys.
**86**(1), 1–48 (1982)zbMATHGoogle Scholar - 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.Aizenman, M., Burchard, A.: Hölder regularity and dimension bounds for random curves. Duke Math. J.
**99**(3), 419–453 (1999)zbMATHGoogle Scholar - 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.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.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.Aizenman, A., Fernández, Roberto: Critical exponents for long-range interactions. Lett. Math. Phys.
**16**, 39–49 (1988)zbMATHGoogle Scholar - 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.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.Benoist, S., Hongler, C.: The scaling limit of critical Ising interfaces is CLE(3) (2016). arXiv:1604.06975
- 11.Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys.
**121**, 501–505 (1989)zbMATHGoogle Scholar - 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.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.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.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.Duminil-Copin, H.: Lectures on the Ising and Potts models on the hypercubic lattice (2017). arXiv:1707.00520
- 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.Duminil-Copin, H., Lis, M.: On the double random current nesting field (2017). arXiv:1712.02305
- 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.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.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.Fisher, M.E.: On the dimer solution of planar Ising models. J. Math. Phys.
**7**(10), 1776–1781 (1966)Google Scholar - 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.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.Griffiths, R.: Correlation in Ising ferromagnets I, II. J. Math. Phys.
**8**, 478–489 (1967)Google Scholar - 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.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.Grimmett, G.: The Random-Cluster Model, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 333. Springer, Berlin (2006)Google Scholar
- 29.Grimmett, G., Janson, S.: Random even subgraphs. Electron. J. Probab.
**16**(1), 1–19 (2009)zbMATHGoogle Scholar - 30.Hegerfeldt, G.C.: Correlation inequalities for Ising ferromagnets with symmetries. Commun. Math. Phys.
**57**(3), 259–266 (1977)Google Scholar - 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.Kadanoff, L.P.: Spin–spin correlation in the two-dimensional Ising model. Nuovo Cimento
**44**, 276–305 (1966)Google Scholar - 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.Kasteleyn, P.W.: Dimer statistics and phase transitions. J. Math. Phys.
**4**, 287–293 (1963)Google Scholar - 35.Kaufman, B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev.
**76**, 1232–1243 (1949)zbMATHGoogle Scholar - 36.Lieb, E.H.: A refinement of Simon’s correlation inequality. Commun. Math. Phys.
**77**, 127–135 (1980)Google Scholar - 37.Lis, M.: The planar Ising model and total positivity. J. Stat. Phys.
**166**(1), 72–89 (2017)zbMATHGoogle Scholar - 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.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.McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)zbMATHGoogle Scholar
- 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.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.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.Pinson, H., Spencer, T.: Universality and the two-dimensional Ising model
**(unpublished preprint)**Google Scholar - 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.Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
**43**, 39–48 (1978)zbMATHGoogle Scholar - 47.Schrader, R.: New correlation inequalities for the Ising model and \(P(\phi )\) theories. Phys. Rev. B
**15**, 2798 (1977)Google Scholar - 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.Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Ann. Discrete Math.
**3**, 227–245 (1978)zbMATHGoogle Scholar - 50.Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys.
**77**, 111–126 (1980)Google Scholar - 51.Tassion, V.: Crossing probabilities for Voronoi percolation. Ann. Probab.
**44**(5), 3385–3398 (2016)zbMATHGoogle Scholar - 52.van der Waerden, B.L.: Die lange Reichweite der regelmassigen Atomanordnung in Mischkristallen. Z. Physik
**118**, 473–488 (1941)zbMATHGoogle Scholar - 53.Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev.
**2**(85), 808–816 (1952)zbMATHGoogle Scholar