Skip to main content

On planar graphs of uniform polynomial growth

Abstract

Consider an infinite planar graph with uniform polynomial growth of degree \(d > 2\). Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of counterexamples. In particular, we show that for every rational \(d > 2\), there is a planar graph with uniform polynomial growth of degree d on which the random walk is transient, disproving a conjecture of Benjamini (Coarse Geometry and Randomness, Volume 2100 of Lecture Notes in Mathematics. Springer, Cham, 2011). By a well-known theorem of Benjamini and Schramm, such a graph cannot be a unimodular random graph. We also give examples of unimodular random planar graphs of uniform polynomial growth with unexpected properties. For instance, graphs of (almost sure) uniform polynomial growth of every rational degree \(d > 2\) for which the speed exponent of the walk is larger than 1/d, and in which the complements of all balls are connected. This resolves negatively two questions of Benjamini and Papasoglou (Proc Am Math Soc 139(11):4105–4111, 2011).

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. More precisely, for the boundary of a hyperbolic group as above, one can choose a sequence of approximations with this property.

References

  1. Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum geometry. A statistical field theory approach. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  2. Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12(54), 1454–1508 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Angel, O.: Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13(5), 935–974 (2003)

    Article  MathSciNet  Google Scholar 

  4. Barlow, M.T.: Diffusions on fractals. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1995), volume 1690 of Lecture Notes in Mathematics. pp. 1–121. Springer, Berlin (1998)

  5. Benjamini, I., Curien, N.: Ergodic theory on stationary random graphs. Electron. J. Probab. 17(93), 20 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Benjamini, I., Curien, N.: Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points. Geom. Funct. Anal. 23(2), 501–531 (2013)

    Article  MathSciNet  Google Scholar 

  7. Benjamini, I.: Coarse geometry and randomness, volume 2100 of Lecture Notes in Mathematics. Springer, Cham, 2013. Lecture notes from the 41st Probability Summer School held in Saint-Flour. Chapter 5 is due to Nicolas Curien, Chapter 12 was written by Ariel Yadin, and Chapter 13 is joint work with Gady Kozma, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School] (2011)

  8. Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150(1), 127–183 (2002)

    Article  MathSciNet  Google Scholar 

  9. Benjamini, I., Papasoglu, P.: Growth and isoperimetric profile of planar graphs. Proc. Am. Math. Soc. 139(11), 4105–4111 (2011)

    Article  MathSciNet  Google Scholar 

  10. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 13 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R., Tiwari, P.: The electrical resistance of a graph captures its commute and cover times. Comput. Complex. 6(4), 312–340 (1996/97)

  12. Gurel-Gurevich, O., Nachmias, A.: Recurrence of planar graph limits. Ann. Math. (2) 177(2), 761–781 (2013)

    Article  MathSciNet  Google Scholar 

  13. Kumagai, T., Misumi, J.: Heat kernel estimates for strongly recurrent random walk on random media. J. Theoret. Probab. 21(4), 910–935 (2008)

    Article  MathSciNet  Google Scholar 

  14. Lee, J.R.: Conformal growth rates and spectral geometry on distributional limits of graphs. (2017). arXiv:1701.01598

  15. Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, New York, (2016). http://pages.iu.edu/~rdlyons/

  16. Murugan, M.: Quasisymmetric uniformization and heat kernel estimates. Trans. Am. Math. Soc. 372(6), 4177–4209 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Shayan Oveis Gharan and Austin Stromme for many useful preliminary discussions, Omer Angel and Asaf Nachmias for sharing with us their construction of a graph with asymptotic \((3-\varepsilon )\)-dimensional volume growth on which the random walk has diffusive speed, and Itai Benjamini for emphasizing many of the questions addressed here. We also thank the anonymous referees for very useful comments. This research was partially supported by NSF CCF-1616297 and a Simons Investigator Award.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to James R. Lee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ebrahimnejad, F., Lee, J.R. On planar graphs of uniform polynomial growth. Probab. Theory Relat. Fields 180, 955–984 (2021). https://doi.org/10.1007/s00440-021-01045-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-021-01045-5

Mathematics Subject Classification

  • 05C10
  • 05B40
  • 05C90
  • 52C26
  • 60D05