On the uniqueness of the infinite cluster of the vacant set of random interlacements
Augusto Teixeira
Source: Ann. Appl. Probab. Volume 19, Number 1
(2009), 454-466.
Abstract
We consider the model of random interlacements on ℤd introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in u of the probability that the origin belongs to the infinite component of the vacant set at level u in the supercritical phase u<u*.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1235140345
Digital Object Identifier: doi:10.1214/08-AAP547
Mathematical Reviews number (MathSciNet): MR2498684
Zentralblatt MATH identifier: 1158.60046
References
[1] Benjamini, I. and Sznitman, A. S. (2008). Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10 1–40.
Mathematical Reviews (MathSciNet): MR2349899
Zentralblatt MATH: 1141.60057
Digital Object Identifier: doi:10.4171/JEMS/106
[2] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
Mathematical Reviews (MathSciNet): MR990777
Zentralblatt MATH: 0662.60113
Digital Object Identifier: doi:10.1007/BF01217735
Project Euclid: euclid.cmp/1104178143
[3] 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.
Mathematical Reviews (MathSciNet): MR2240791
Zentralblatt MATH: 1105.60029
Digital Object Identifier: doi:10.1007/s00440-005-0485-9
[4] Häggström, O. and Jonasson, J. (2006). Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 289–344.
Mathematical Reviews (MathSciNet): MR2280297
Digital Object Identifier: doi:10.1214/154957806000000096
Project Euclid: euclid.ps/1167536197
[5] Häggström, O. and Peres, Y. (1999). Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultaneously. Probab. Theory Related Fields 113 273–285.
Mathematical Reviews (MathSciNet): MR1676835
Zentralblatt MATH: 0921.60091
Digital Object Identifier: doi:10.1007/s004400050208
[6] Newman, C. M. and Schulman, L. S. (1981). Infinite clusters in percolation models. J. Statist. Phys. 26 613–628.
Mathematical Reviews (MathSciNet): MR648202
Zentralblatt MATH: 0509.60095
Digital Object Identifier: doi:10.1007/BF01011437
[7] Sidoravicius, V. and Sznitman, A. S. (2008). Percolation for the vacant set of random interlacements. Preprint. Available at http://www.math.ethz.ch/u/sznitman/preprints/.
Mathematical Reviews (MathSciNet): MR2512613
Zentralblatt MATH: 1168.60036
Digital Object Identifier: doi:10.1002/cpa.20267
[8] Sznitman, A. S. (2007). Vacant set of random interlacements and percolation. Preprint. Available at http://www.math.ethz.ch/u/sznitman/preprints/.
[9] Sznitman, A. S. (2007). Random walks on discrete cylinders and random interlacements. Preprint. Available at http://www.math.ethz.ch/u/sznitman/preprints/.
Mathematical Reviews (MathSciNet): MR2520124
Zentralblatt MATH: 1172.60316
Digital Object Identifier: doi:10.1007/s00440-008-0164-8
[10] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
Mathematical Reviews (MathSciNet): MR900810
[11] van den Berg, J. and Keane, M. (1984). On the continuity of the percolation probability function. In Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Contemporary Mathematicians 26 61–65. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR737388
Zentralblatt MATH: 0541.60099
The Annals of Applied Probability
