Skip to main content

Local Limits of Graphs

  • 1556 Accesses

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2100)

Abstract

In this section we will only consider connected simple graphs (i.e. without loops or multiple edges). We start by recalling few definitions from previous sections. If G = (V, E) is such a graph, and x, yV, the graph distance between x and y in G is defined to be the length of a shortest path in G between x and y, and is denoted by d G (x, y). A rooted graph (G, ρ) is a graph G together with a distinguished vertex ρ of G.

Keywords

  • Vertex-transitive Graphs
  • Simple Random Walk Path
  • Sharp Error Term
  • Measurable Equivalence Relation
  • Mass Transport Principle

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   49.99
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. N. Alon, I. Benjamini, A. Stacey, Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32(3A), 1727–1745 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. I. Benjamini, R. Lyons, Y. Peres, O. Schramm, Group-invariant percolation on graphs. Geom. Funct. Anal. 9(1), 29–66 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. I. Benjamini, A. Nachmias, Y. Peres, Is the critical percolation probability local? Probab. Theory Relat. Fields 149(1–2), 261–269 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  7. J.W. Cannon, W.J. Floyd, R. Kenyon, W.R. Parry, Hyperbolic geometry, in Flavors of Geometry. Mathematical Sciences Research Institute Publications, vol. 31 (Cambridge University Press, Cambridge, 1997), pp. 59–115

    Google Scholar 

  8. G. Grimmett, Percolation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. (Springer, Berlin, 1999)

    Google Scholar 

  9. O. Häggström, Two badly behaved percolation processes on a nonunimodular graph. J. Theor. Probab. 24, 1–16 (2011)

    CrossRef  Google Scholar 

  10. T. Hara, G. Slade, Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128(2), 333–391 (1990)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. R. Lyons, R. Pemantle, Y. Peres, Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure. Ergodic Theory Dyn. Syst. 15(3), 593–619 (1995)

    CrossRef  MathSciNet  Google Scholar 

  12. R. Lyons, O. Schramm, Indistinguishability of percolation clusters. Ann. Probab. 27(4), 1809–1836 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benjamini, I. (2013). Local Limits of Graphs. In: Coarse Geometry and Randomness. Lecture Notes in Mathematics(), vol 2100. Springer, Cham. https://doi.org/10.1007/978-3-319-02576-6_5

Download citation