Upper bound on the disconnection time of discrete cylinders and random interlacements
Alain-Sol Sznitman
Source: Ann. Probab.
Volume 37, Number 5
(2009), 1715-1746.
Abstract
We study the asymptotic behavior for large N of the disconnection time TN of a simple random walk on the discrete cylinder (ℤ/Nℤ)d×ℤ, when d≥2. We explore its connection with the model of random interlacements on ℤd+1 recently introduced in [Ann. Math., in press], and specifically with the percolative properties of the vacant set left by random interlacements. As an application we show that in the large N limit the tail of TN/N2d is dominated by the tail of the first time when the supremum over the space variable of the Brownian local times reaches a certain critical value. As a by-product, we prove the tightness of the laws of TN/N2d, when d≥2.
Primary Subjects: 60G50, 60K35, 82C41
Keywords: Disconnection; random walks; random interlacements; discrete cylinders
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1253539855
Digital Object Identifier: doi:10.1214/09-AOP450
Zentralblatt MATH identifier:
05625051
References
[1] Borodin, A. N. (1982). Distribution of integral functionals of the Brownian motion. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 119 19–38.
Mathematical Reviews (MathSciNet):
MR666084
[2] Csáki, E. and Révész, P. (1983). Strong invariance for local times. Z. Wahrsch. Verw. Gebiete 62 263–278.
[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.
[4] Dembo, A. and Sznitman, A.-S. (2008). A lower bound on the disconnection time of a discrete cylinder. In In and Out of Equilibrium. 2. Progress in Probability 60 211–227. Birkhäuser, Basel.
[5] Eisenbaum, N. (1990). Un théorème de Ray–Knight lié au supremum des temps locaux browniens. Probab. Theory Related Fields 87 79–95.
[6] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
[7] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Basel.
[8] Lindvall, T. (1992). Lectures on the Coupling Method. Dover, New York.
[9] Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.
Mathematical Reviews (MathSciNet):
MR435697
[10] Sidoravicius, V. and Sznitman, A. S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
[11] Sznitman, A.-S. (2008). How universal are asymptotics of disconnection times in discrete cylinders? Ann. Probab. 36 1–53.
[12] Sznitman, A. S. Vacant set of random interlacements and percolation. Ann. Math. To appear. Available at http://www.math.ethz.ch/u/sznitman/preprints.
[13] Sznitman, A.-S. (2009). Random walks on discrete cylinders and random interlacements. Probab. Theory Related Fields 145 143–174.
