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 walks on discrete cylinders and random interlacements
Download PDF
Download PDF
  • Published: 17 July 2008

Random walks on discrete cylinders and random interlacements

  • Alain-Sol Sznitman1 

Probability Theory and Related Fields volume 145, pages 143–174 (2009)Cite this article

  • 147 Accesses

  • 21 Citations

  • Metrics details

Abstract

We explore some of the connections between the local picture left by the trace of simple random walk on a cylinder \({(\mathbb {Z} / N\mathbb {Z})^d \times \mathbb {Z}}\) , d ≥ 2, running for times of order N 2d and the model of random interlacements recently introduced in Sznitman ( http://www.math.ethz.ch/u/sznitman/preprints). In particular, we show that for large N in the neighborhood of a point of the cylinder with vertical component of order N d the complement of the set of points visited by the walk up to times of order N 2d is close in distribution to the law of the vacant set of random interlacements with a level which is determined by an independent Brownian local time. The limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Chung K.L. (1974) A Course in Probability Theory. Academic Press, San Diego

    MATH  Google Scholar 

  2. Csáki E., Revesz P. (1983) Strong invariance for local times. Z. für Wahrsch. verw. Geb. 62: 263–278

    Article  MATH  Google Scholar 

  3. Dembo A., Sznitman A.S. (2006) On the disconnection of a discrete cylinder by a random walk. Probab. Theory Relat. Fields 136(2): 321–340

    Article  MATH  MathSciNet  Google Scholar 

  4. Dembo, A., Sznitman, A.S.: A lower bound on the disconnection time of a discrete cylinder. Progress in probability, vol. 60. In and Out of Equilibrium 2. Birkhäuser, Basel, pp. 211–227 (2008)

  5. Grigoryan A., Telcs A. (2001) Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109(3): 451–510

    Article  MathSciNet  Google Scholar 

  6. Khaśminskii R.Z. (1959) On positive solutions of the equation A u  +  V u  =  0. Theor. Probab. Appl. 4: 309–318

    Article  Google Scholar 

  7. Lindvall T. (1992) Lectures on the Coupling Method. Dover, New York

    MATH  Google Scholar 

  8. Sznitman A.S. (2008) How universal are asymptotics of disconnection times in discrete cylinders? Ann. Probab. 36(1): 1–53

    Article  MATH  MathSciNet  Google Scholar 

  9. Sznitman, A.S.: Vacant set of random interlacements and percolation. Preprint available at: http://www.math.ethz.ch/u/sznitman/preprints

  10. Sznitman, A.S.: Upper bound on the disconnection time of discrete cylinders and random interlacements. Preprint available at: http://www.math.ethz.ch/u/sznitman/preprints

  11. Sidoravicius, V., Sznitman, A.S.: Percolation for the vacant set of random interlacements. Preprint available at: http://www.math.ethz.ch/u/sznitman/preprints

  12. Teixeira, A.: On the uniqueness of the infinite cluster of the vacant set of random interlacements. Ann. Appl. Probab. (in press) Also available at arXiv:0805.4106

  13. Windisch D. (2008) Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13: 140–150

    MathSciNet  Google Scholar 

  14. Woess W. (2000) Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement Mathematik, ETH Zürich, 8092, Zurich, Switzerland

    Alain-Sol Sznitman

Authors
  1. Alain-Sol Sznitman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain-Sol Sznitman.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sznitman, AS. Random walks on discrete cylinders and random interlacements. Probab. Theory Relat. Fields 145, 143–174 (2009). https://doi.org/10.1007/s00440-008-0164-8

Download citation

  • Received: 31 October 2007

  • Revised: 29 May 2008

  • Published: 17 July 2008

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00440-008-0164-8

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

Mathematics Subject Classification (2000)

  • 60G50
  • 60K35
  • 82C41
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

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature