Skip to main content

Gaussian Free Field on Hyperbolic Lattices

Part of the Lecture Notes in Mathematics book series (LNM,volume 2116)

Abstract

It is shown that the maximum of the height of the Gaussian free field in a ball of a two dimensional hyperbolic lattice, grows linearly with the radius, while only as a square root of the radius on higher dimensional hyperbolic lattices.

Keywords

  • Gaussian Free Field (GFF)
  • Hyperbolic Lattice
  • Random Lipschitz Functions
  • Euclidean Lattice
  • Exponential Volume Growth

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   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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
Fig. 1
Fig. 2

References

  1. I. Benjamini, O. Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Geom. Funct. Anal. 7, 403–419 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. I. Benjamini, R. Lyons, Y. Peres, O. Schramm, Uniform spanning forests. Ann. Probab. 29, 1–65 (2001)

    MathSciNet  MATH  Google Scholar 

  3. J. Canon, W. Floyed, R. Kenyon, W. Parry, Hyperbolic geometry. Math. Sci. Res. Inst. Publ. 31, 59–115 (1997)

    Google Scholar 

  4. P. Doyle, S. Snell, Random Walks and Electric Networks (Mathematical Association of America, America, 1984)

    MATH  Google Scholar 

  5. X. Fernique, Regularite des trajectoires des fonctions aleatoires gaussiennes. Lecture Notes in Mathematics, vol. 480 (Springer, Berlin, 1975)

    Google Scholar 

  6. S. Janson, Gaussian Hilbert Spaces (Cambridge University Press, Cambridge, 1998)

    Google Scholar 

  7. R. Lyons, Y. Peres, Probability on Trees and Networks (Cambridge University Press, Cambridge, 2013). Current version available at http://mypage.iu.edu/~rdlyons/

  8. R. Peled, High-dimensional Lipschitz functions are typically flat. Ann. Probab. (to appear). http://arxiv.org/abs/1005.4636

  9. Y. Velenik, Localization and delocalization of random interfaces. Probab. Surv. 3, 112–169 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Thanks to Omer Angel, Alessandro Carderi and Yuval Peres. Sergey Khristo helped with the writing and figures.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Itai Benjamini .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benjamini, I. (2014). Gaussian Free Field on Hyperbolic Lattices. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_2

Download citation