Skip to main content
SpringerLink
Log in

Decoupling inequalities and interlacement percolation on G×ℤ

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We study the percolative properties of random interlacements on G×ℤ, where G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters α>1 and 2≤β≤α+1, describing the respective polynomial growths of the volume on G and of the time needed by the walk on G to move to a distance. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when α≥1+β/2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where G=ℤd, d≥2, several of these results are new.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abete, T., de Candia, A., Lairez, D., Coniglio, A.: Percolation model for enzyme gel degradation. Phys. Rev. Lett. 93, 228301 (2004)

    Article  Google Scholar 

  2. Barlow, M.T.: Which values of the volume growth and escape time exponent are possible for a graph? Rev. Mat. Iberoam. 20(1), 1–31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barlow, M.T., Coulhon, T., Kumagai, T.: Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Commun. Pure Appl. Math. 58(12), 1642–1677 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belius, D.: Cover times in the discrete cylinder. Available at arXiv:1103.2079

  5. Benjamini, I., Sznitman, A.S.: Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10(1), 133–172 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Černý, J., Teixeira, A.: Critical window for the vacant set left by random walk on random regular graphs. Available at arXiv:1101.1978

  7. Černý, J., Teixeira, A., Windisch, D.: Giant vacant component left by a random walk in a random d-regular graph. To appear in Ann. Inst. Henri Poincaré, also available at arXiv:1012.5117

  8. Chung, K.L., Zhao, Z.Z.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  9. Cooper, C., Frieze, A.: Component structure induced by a random walk on a random graph. Available at arXiv:1005.1564

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Grigoryan, A., Telcs, A.: Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324(3), 521–556 (2002)

    Article  MathSciNet  Google Scholar 

  13. Grimmett, G.: Percolation., 2nd edn. Springer, Berlin (1999)

    Google Scholar 

  14. Hambly, B.M., Kumagai, T.: Heat kernel estimates for symmetric random walks in a class of fractal graphs and stability under rough isometries. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Proc. Symp. Pure Math. 72(2), 233–259 (2004)

    MathSciNet  Google Scholar 

  15. Jones, O.D.: Transition probabilities for the simple random walk on the Sierpinski graph. Stoch. Process. Appl. 61(1), 45–69 (1996)

    Article  MATH  Google Scholar 

  16. Khaśminskii, R.Z.: On positive solutions of the equation Au+Vu=0. Theory Probab. Appl. 4, 309–318 (1959)

    Article  Google Scholar 

  17. Kumagai, T.: Random walks on disordered media and their scaling limits. Notes of St. Flour lectures, also available at http://www.kurims.kyoto-u.ac.jp/~kumagai/StFlour-Cornell.html (2010)

  18. Sidoravicius, V., Sznitman, A.S.: Percolation for the vacant set of random interlacements. Commun. Pure Appl. Math. 62(6), 831–858 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sidoravicius, V., Sznitman, A.S.: Connectivity bounds for the vacant set of random interlacements. Ann. Inst. H. Poincaré 46(4), 976–990 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Sznitman, A.S.: Vacant set of random interlacements and percolation. Ann. Math. 171, 2039–2087 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sznitman, A.S.: Random walks on discrete cylinders and random interlacements. Probab. Theory Relat. Fields 145, 143–174 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sznitman, A.S.: Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37(5), 1715–1746 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sznitman, A.S.: On the domination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab. 14, 1670–1704 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sznitman, A.S.: On the critical parameter of interlacement percolation in high dimension. Ann. Probab. 39(1), 70–103 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Teixeira, A.: Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14, 1604–1627 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Teixeira, A.: On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Relat. Fields, doi:10.1007/s00440-010-0283-x, also available at arXiv:1002.4995

  28. Teixeira, A., Windisch, D.: On the fragmentation of a torus by random walk. To appear in Comm. Pure Appl. Math., also available at arXiv:1007.0902

  29. Watkins, M.E.: Infinite paths that contain only shortest paths. J. Comb. Theory, Ser. B, 41:341–355 (1986)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Windisch, D.: Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. 38(2), 841–895 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement Mathematik, ETH-Zentrum, 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. Decoupling inequalities and interlacement percolation on G×ℤ. Invent. math. 187, 645–706 (2012). https://doi.org/10.1007/s00222-011-0340-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-011-0340-9

Keywords

Search

Navigation