Skip to main content
SpringerLink
Log in

Lace Expansion for the Ising Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with \({d\,\gg\,4}\) and for the spread-out model with d  >  4 and \({L\,\gg\,1}\) , without assuming reflection positivity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman M. (1982). Geometric analysis of \({\phi^4}\) fields and Ising models. Commun. Math. Phys. 86: 1–48

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Aizenman M., Barsky D.J. and Fernández R. (1987). The phase transition in a general class of Ising-type models is sharp. J. Statist. Phys. 47: 343–374

    Article  MathSciNet  ADS  Google Scholar 

  3. Aizenman M. and Fernández R. (1986). On the critical behavior of the magnetization in high-dimensional Ising models. J. Statist. Phys. 44: 393–454

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Aizenman M. and Graham R. (1983). On the renormalized coupling constant and the susceptibility in \({\phi_4^4}\) field theory and the Ising model in four dimensions. Nucl. Phys. B 225: 261–288

    Article  ADS  MathSciNet  Google Scholar 

  5. Kesten H. and Berg J. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22: 556–569

    Article  MATH  MathSciNet  Google Scholar 

  6. Biskup M., Chayes L. and Crawford N. (2006). Mean-field driven first-order phase transitions in systems with long-range interactions. J. Stat. Phys. 122: 1139–1193

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Bodineau T. (2006). Translation invariant Gibbs states for the Ising model. Probab. Th. Rel. Fields. 135: 153–168

    Article  MATH  MathSciNet  Google Scholar 

  8. Brydges D. and Spencer T. (1985). Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys. 97: 125–148

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Campanino M., Ioffe D. and Velenik Y. (2003). Ornstein-Zernike theory for finite range Ising models above T c . Probab. Th. Rel. Fields 125: 305–349

    Article  MATH  MathSciNet  Google Scholar 

  10. Fernández R., Fröhlich J. and Sokal A.D. (1992). Random walks, critical phenomena, and triviality in quantum field theory. Springer, Berlin

    MATH  Google Scholar 

  11. Fortuin C.M. and Kasteleyn P.W. (1972). On the random-cluster model I. Introduction and relation to other models. Physica 57: 536–564

    Article  ADS  MathSciNet  Google Scholar 

  12. Fröhlich J., Simon B. and Spencer T. (1976). Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys. 50: 79–95

    Article  ADS  Google Scholar 

  13. Griffiths R.B., Hurst C.A. and Sherman S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11: 790–795

    Article  MathSciNet  ADS  Google Scholar 

  14. Hara, T.: Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. http://arivx.org/list/math-ph/0504021, 2005

  15. Hara T., van der Hofstad R. and Slade G. (2003). Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31: 349–408

    Article  MATH  MathSciNet  Google Scholar 

  16. Hara T. and Slade G. (1990). On the upper critical dimension of lattice trees and lattice animals. J. Statist. Phys. 59: 1469–1510

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Hara T. and Slade G. (1990). Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128: 333–391

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Hara T. and Slade G. (1992). Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys. 147: 101–136

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. Hara, T., Slade, G.: Mean-field behaviour and the lace expansion. In: Probability and Phase Transition, ed., G. R. Grimmett, Dordrecht: Kluwer, pp. 87–122

  20. Heydenreich, M., van der Hofstad, R., Sakai, A.: Mean-field behavior for long- and finite-range Ising model, percolation and self-avoiding walk. In preparation

  21. van der Hofstad R. and Slade G. (2002). A generalised inductive approach to the lace expansion. Probab. Th. Rel. Fields 122: 389–430

    Article  MATH  Google Scholar 

  22. Ising E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31: 253–258

    Article  ADS  Google Scholar 

  23. Lieb E.H. (1980). A refinement of Simon’s correlation inequality. Commun. Math. Phys. 77: 127–135

    Article  ADS  MathSciNet  Google Scholar 

  24. Madras N. and Slade G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston

    MATH  Google Scholar 

  25. Nguyen B.G. and Yang W-S. (1993). Triangle condition for oriented percolation in high dimensions. Ann. Probab. 21: 1809–1844

    MATH  MathSciNet  Google Scholar 

  26. Onsager L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65: 117–149

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Sakai A. (2001). Mean-field critical behavior for the contact process. J. Statist. Phys. 104: 111–143

    Article  MATH  MathSciNet  Google Scholar 

  28. Sakai, A.: Asymptotic behavior of the critical two-point function for spread-out Ising ferromagnets above four dimensions. Unpublished manuscript, (2005)

  29. Sakai, A.: Diagrammatic estimates of the lace-expansion coefficients for finite-range Ising ferromagnets. Unpublished notes, (2006)

  30. Simon B. (1980). Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys. 77: 111–126

    Article  ADS  Google Scholar 

  31. Slade, G.: The lace expansion and its applications. Springer Lecture Notes in Mathematics 1879, Berlin Heidelberg-New York: Springer, 2006

  32. Sokal A.D. (1982). An alternate constructive approach to the \({\varphi_3^4}\) quantum field theory, and a possible destructive approach to \({\varphi_4^4}\). Ann. Inst. Henri Poincaré Phys. Théorique 37: 317–398

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Akira Sakai

Authors
  1. Akira Sakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akira Sakai.

Additional information

Communicated by J.Z. Imbrie.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sakai, A. Lace Expansion for the Ising Model. Commun. Math. Phys. 272, 283–344 (2007). https://doi.org/10.1007/s00220-007-0227-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0227-1

Keywords

Search

Navigation