Home | Contents | Submissions, editors, etc. | Login | Search | ECP
 Electronic Journal of Probability > Vol. 14 (2009) > Paper 54 open journal systems 


Interlacement percolation on transient weighted graphs

Augusto Teixeira, Eidgenössische Technische Hochschule Zürich


Abstract
In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u* is positive for the product GxZ (where we endow Z with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value u*.


Full text: PDF

Pages: 1604-1627

Published on: July 9, 2009


Bibliography
  1. N. Alon, I. Benjamini, A. Stacey. Percolation on finite graphs and isoperimetric inequalities. Ann. Probab., 32, (2004) 3, 1727-1745. Math. Review 2005f:05149
  2. I. Benjamini, A.S. Sznitman. Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc., 10 (2008) 1, 1-40. Math. Review 2008i:60076
  3. C.M. Fortuin, P.W. Kasteleyn, J. Ginibre. Correlation inequalities on some partially ordered sets. Comm. Math. Phys., 22 (1971) 89-103. Math. Review 46 #8607
  4. A. Dembo, A.S. Sznitman. On the disconnection of a discrete cylinder by a random walk. Probab. Theory Relat. Fields, 136 (2006) 2, 321-340. Math. Review 2007i:60053
  5. G. Grimmett. Percolation, Springer Verlag. second edition (1999). Math. Review 2001a:60114
  6. G. Grimmett, D. Stirzaker. Probability and random processes. Clarendon Press, Oxford, second edition (1992). Math. Review 93m:60002
  7. G.A. Hunt. Markoff chains and Martin boundaries. Illinoiws J. Math., 4 (1960) 313-340. Math. Review 23 #A691
  8. H. Kesten. Asymptotics in high dimensions for percolation. Disorder in physical systems: A volume in honour of John Hammersley (ed. G. Grimmett and D. J. A. Welsh), Clarendon Press, Oxford (1990) 219-240. Math. Review 91k:60114
  9. G.F. Lawler. Intersections of random walks. Birkhäuser (1991). Math. Review 92f:60122
  10. T.M. Liggett. Interacting Particle Systems. Springer Verlag (2005). Math. Review 2006b:60003
  11. V. Sidoravicius, A.S. Sznitman. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math., 62 (6) (2009), 831-858. Math. Review number not available.
  12. M.L. Silverstein. Symmetric Markov process. Lecture Notes in Math. 426, Springer Verlag, (1974). Math. Review 52 #6891
  13. A.S. Sznitman. How universal are asymptotics of disconnection times in discrete cylinders. Ann. Probab., 36 (2008) 1, 1-53. Math. Review 2008m:60143
  14. A.S. Sznitman. Vacant set of random interlacements and percolation. Accepted for publication in the Annals of Mathematics. Preprint available at www.math.ethz.ch/u/sznitman/ (2007).
  15. A.S. Sznitman. Random walks on discrete cylinders and random interlacements. Probab. Theory Relat. Fields, 145 (2009) 143-174. Math. Review number not available.
  16. A.S. Sznitman. Upper bound on the disconnection time of discrete cylinders and random interlacements . Accepted for publication in the Annals of Probability. Preprint available at www.math.ethz.ch/u/sznitman/ (2008).
  17. S.I. Resnick. Extreme Values, regular variation and point processes. Springer Verlag, (1987). Math. Review 89b:60241
  18. M. Watkins. Infinite paths that contain only shortest paths. Journ. of Combinat. Theory, 41 (1986) 341-355. Math. Review 87m:05118
  19. M. Weil. Quasi-processus. Séminaire de Probabilités IV, Lecture Notes in Math. 124, Springer Verlag, 217-239 (1970). Math. Review 42 #1212
  20. D. Windisch. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008), 140-150. Math. Review 2008k:60249
  21. W. Woess. Random walks on infinite graphs and groups. Cambridge University Press (2000). Math. Review 2001k:60006
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | ECP

Electronic Journal of Probability. ISSN: 1083-6489