Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time
Download PDF
Download PDF
  • Open Access
  • Published: 11 December 2009

Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time

  • Markus Heydenreich1 &
  • Remco van der Hofstad2 

Probability Theory and Related Fields volume 149, pages 397–415 (2011)Cite this article

  • 581 Accesses

  • 21 Citations

  • Metrics details

A Correction to this article was published on 09 August 2019

This article has been updated

Abstract

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335–358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551–582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267–1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdős-Rényi random graphs.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

Change history

  • 09 August 2019

    In [3, Theorem 1.2], we claim that the maximal cluster for critical percolation on the high-dimensional torus is non-concentrated. This proof contains an error. In this note, we replace this statement by a conditional statement instead.

  • 09 August 2019

    In [3, Theorem 1.2], we claim that the maximal cluster for critical percolation on the high-dimensional torus is non-concentrated. This proof contains an error. In this note, we replace this statement by a conditional statement instead.

References

  1. Aizenman M.: On the number of incipient spanning clusters. Nuclear. Phys. B 485(3), 551–582 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aizenman M., Barsky D.J.: Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108(3), 489–526 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aizenman M., Newman C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36(1–2), 107–143 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aldous D.: Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25(2), 812–854 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Borgs C., Chayes J.T., van der Hofstad R., Slade G., Spencer J.: Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Struct. Algorithms 27(2), 137–184 (2005)

    Article  MATH  Google Scholar 

  7. Borgs C., Chayes J.T., van der Hofstad R., Slade G., Spencer J.: Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33(5), 1886–1944 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. Hara T.: Mean-field critical behaviour for correlation length for percolation in high dimensions. Probab. Theory Relat. Fields 86(3), 337–385 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hara T.: Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36(2), 530–593 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hara T., van der Hofstad R., Slade G.: Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31(1), 349–408 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Hara T., Slade G.: The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys. 99(5–6), 1075–1168 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hara T., Slade G.: The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41(3), 1244–1293 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Heydenreich M., van der Hofstad R.: Random graph asymptotics on high-dimensional tori. Comm. Math. Phys. 270(2), 335–358 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kozma G., Nachmias A.: The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178(3), 635–654 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Menshikov M.V.: Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288(6), 1308–1311 (1986)

    MathSciNet  Google Scholar 

  18. Nachmias, A., Peres, Y.: Critical percolation on random regular graphs. Preprint arXiv:0707.2839v2 [math.PR], 2007. To appear in Random Structures and Algorithms

  19. Nachmias A., Peres Y.: Critical random graphs: diameter and mixing time. Ann. Probab. 36(4), 1267–1286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Slade, G.: The lace expansion and its applications. Lecture Notes in Mathematics, vol. 1879, xiv+232 pp. Springer, Berlin (2006)

Download references

Acknowledgements

The work of RvdH was supported in part by the Netherlands Organisation for Scientific Research (NWO). We thank Asaf Nachmias for enlightening discussions concerning the results and methodology in [16,19]. MH is grateful to Institut Mittag-Leffler for the kind hospitality during his stay in February 2009, and in particular to Jeff Steif for inspiring discussions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    Markus Heydenreich

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Remco van der Hofstad

Authors
  1. Markus Heydenreich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Remco van der Hofstad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Remco van der Hofstad.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Heydenreich, M., van der Hofstad, R. Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time. Probab. Theory Relat. Fields 149, 397–415 (2011). https://doi.org/10.1007/s00440-009-0258-y

Download citation

  • Received: 20 April 2009

  • Revised: 09 November 2009

  • Published: 11 December 2009

  • Issue Date: April 2011

  • DOI: https://doi.org/10.1007/s00440-009-0258-y

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Percolation
  • Random graph asymptotics
  • Mean-field behavior
  • Critical window

Mathematics Subject Classification (2000)

  • 60K35
  • 82B43
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature