Skip to main content

Alexander–Orbach Conjecture Holds When Two-Point Functions Behave Nicely

  • 1484 Accesses

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2101)

Abstract

This chapter is based on the paper by Kozma-Nachmias [162] with some simplification by [197]. Our framework here is unimodular transitive graphs that contains \({\mathbb{Z}}^{d}\) as a typical example.

Keywords

  • Critical Bond Percolation
  • Lace Expansion
  • Subcritical Branching Processes
  • Nearest-neighbor Model
  • Graph Distance

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   44.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. M. Aizenman, D.J. Barsky, Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526 (1987)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. M. Aizenman, C.M. Newman, Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36, 107–143 (1984)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. M.T. Barlow, T. Kumagai, Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50, 33–65 (2006) (electronic)

    Google Scholar 

  4. D.J. Barsky, M. Aizenman, Percolation critical exponents under the triangle condition. Ann. Probab. 19, 1520–1536 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. B. Bollobás, Random Graphs, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    CrossRef  MATH  Google Scholar 

  6. Z.-Q. Chen, P. Kim, T. Kumagai, Discrete approximation of symmetric jump processes on metric measure spaces. Probab. Theory Relat. Fields 155, 703–749 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. N. Crawford, A. Sly, Simple random walks on long range percolation clusters I: heat kernel bounds. Probab. Theory Relat. Fields 154, 753–786 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. N. Crawford, A. Sly, Simple random walks on long range percolation clusters II: scaling limits. Ann. Probab. 41, 445–502 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. G. Grimmett, Percolation, 2nd edn. (Springer, Berlin, 1999)

    CrossRef  MATH  Google Scholar 

  10. T. Hara, Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36, 530–593 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. M. Heydenreich, R. van der Hofstad, T. Hulshof, Random walk on the high-dimensional IIC. ArXiv:1207.7230 (2012)

    Google Scholar 

  13. R. van der Hofstad, A.A. Járai, The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114, 625–663 (2004)

    CrossRef  MATH  Google Scholar 

  14. G. Kozma, Percolation on a product of two trees. Ann. Probab. 39, 1864–1895 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. G. Kozma, The triangle and the open triangle. Ann. Inst. Henri Poincaré Probab. Stat. 47, 75–79 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. G. Kozma, A. Nachmias, The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178, 635–654 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. G. Kozma, A. Nachmias, Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24, 375–409 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. A. Nachmias, Y. Peres, Critical random graphs: diameter and mixing time. Ann. Probab. 36, 1267–1286 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. A. Sapozhnikov, Upper bound on the expected size of intrinsic ball. Electron. Commun. Probab. 15, 297–298 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. R. Schonmann, Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219, 271–322 (2001)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kumagai, T. (2014). Alexander–Orbach Conjecture Holds When Two-Point Functions Behave Nicely. In: Random Walks on Disordered Media and their Scaling Limits. Lecture Notes in Mathematics(), vol 2101. Springer, Cham. https://doi.org/10.1007/978-3-319-03152-1_6

Download citation