The Annals of Applied Probability

On the disconnection of a discrete cylinder by a biased random walk

David Windisch

Full-text: Open access

Abstract

We consider a random walk on the discrete cylinder (ℤ/Nℤ)d×ℤ, d≥3 with drift N in the ℤ-direction and investigate the large N-behavior of the disconnection time TNdisc, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent α is strictly greater than 1, the asymptotic behavior of TNdisc remains N2d+o(1), as in the unbiased case considered by Dembo and Sznitman, whereas for α<1, the asymptotic behavior of TNdisc becomes exponential in N.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 4 (2008), 1441-1490.

Dates
First available in Project Euclid: 21 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1216677128

Digital Object Identifier
doi:10.1214/07-AAP491

Mathematical Reviews number (MathSciNet)
MR2434177

Zentralblatt MATH identifier
1148.60028

Subjects
Primary: 60G50: Sums of independent random variables; random walks

Keywords
Random walk discrete cylinder disconnection

Citation

Windisch, David. On the disconnection of a discrete cylinder by a biased random walk. Ann. Appl. Probab. 18 (2008), no. 4, 1441--1490. doi:10.1214/07-AAP491. https://projecteuclid.org/euclid.aoap/1216677128


Export citation

References

  • [1] Aldous, D. J. and Fill, J. (2008). Reversible Markov Chains and Random Walks on Graphs. Available at http://www.stat.Berkeley.EDV/users/aldous/book.html.
  • [2] Alon, N., Spencer, J. H. and Erdős, P. (1992). The Probabilistic Method. Wiley, New York.
  • [3] Benjamini, I. and Sznitman, A. S. (2008). Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. (JEMS) 10 133–172.
  • [4] Dembo, A. and Sznitman, A. S. (2006). On the disconnection of a discrete cylinder by a random walk. Probab. Theory Related Fields 136 321–340.
  • [5] Dembo, A. and Sznitman, A. S. (2008). A lower bound on the disconnection time of a discrete cylinder. Preprint.
  • [6] Deuschel, J. D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 467–482.
  • [7] Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Brooks/Cole, Belmont, CA.
  • [8] Khaśminskii, R. Z. (1959). On positive solutions of the equation U+Vu=0. Theory Probab. Appl. 4 309–318.
  • [9] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Basel.
  • [10] Loomis, L. H. and Whitney, H. (1949). An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55 961–962.
  • [11] Sznitman, A. S. (2003). On new examples of ballistic random walks in random environment. Ann. Probab. 31 285–322.
  • [12] Sznitman, A. S. (2008). How universal are asymptotics of disconnection times in discrete cylinders? Ann. Probab. 36 1–53.
  • [13] Sznitman, A. S. (2007). Vacant set of random interlacements and percolation. Preprint. Available at http://arxiv.org/abs/0704.2560.