The Annals of Probability

Random walks on discrete cylinders with large bases and random interlacements

David Windisch

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Following the recent work of Sznitman [Probab. Theory Related Fields 145 (2009) 143–174], we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form GN×ℤ, where GN is a large finite connected weighted graph, and relate it to the model of random interlacements on infinite transient weighted graphs. Under suitable assumptions, the set of points not visited by the random walk until a time of order |GN|2 in a neighborhood of a point with ℤ-component of order |GN| converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point. The level of the random interlacement depends on the local time of a Brownian motion. The result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models. As examples of GN, we treat the d-dimensional box of side length N, the Sierpinski graph of depth N and the d-ary tree of depth N, where d≥2.

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Ann. Probab., Volume 38, Number 2 (2010), 841-895.

First available in Project Euclid: 9 March 2010

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Random walk discrete cylinder random interlacements


Windisch, David. Random walks on discrete cylinders with large bases and random interlacements. Ann. Probab. 38 (2010), no. 2, 841--895. doi:10.1214/09-AOP497.

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  • [1] Aldous, D. J. and Fill, J. (2002). Reversible Markov chains and random walks on graphs. Available at http://www.stat.Berkeley.EDU/users/aldous/book.html.
  • [2] Barlow, M. T., Coulhon, T. and Kumagai, T. (2005). Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58 1642–1677.
  • [3] Chung, K. L. (1974). A Course in Probability Theory, 2nd ed. Academic Press, New York.
  • [4] Darling, R. W. R. and Norris, J. R. (2008). Differential equation approximations for Markov chains. Probab. Surv. 5 37–79.
  • [5] Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Brooks/Cole, Belmont.
  • [6] Chou, C. S. and Meyer, P. A. (1975). Sur la représentation des martingales comme intégrales stochastiques dans les processus ponctuels. In Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, Années Universitaires 1973/1974 et 1974/1975). Lecture Notes in Math. 465 226–236. Springer, Berlin.
  • [7] Jones, O. D. (1996). Transition probabilities for the simple random walk on the Sierpiński graph. Stochastic Process. Appl. 61 45–69.
  • [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, Boston, MA.
  • [10] Norris, J. R. (1997). Markov Chains. Cambridge Univ. Press, New York.
  • [11] Révész, P. (1981). Local time and invariance. In Analytical Methods in Probability Theory (Oberwolfach, 1980). Lecture Notes in Math. 861 128–145. Springer, Berlin.
  • [12] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301–413. Springer, Berlin.
  • [13] Shima, T. (1991). On eigenvalue problems for the random walks on the Sierpiński pre-gaskets. Japan J. Indust. Appl. Math. 8 127–141.
  • [14] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
  • [15] Sznitman, A.-S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115 287–323.
  • [16] Sznitman, A. S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2). To appear. Available at
  • [17] Sznitman, A.-S. (2009). Random walks on discrete cylinders and random interlacements. Probab. Theory Related Fields 145 143–174.
  • [18] Sznitman, A. S. (2009). Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37 1715–1746.
  • [19] Teixeira, A. (2009). Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 no. 54, 1604–1628.
  • [20] Telcs, A. (2006). The Art of Random Walks. Lecture Notes in Math. 1885. Springer, Berlin.
  • [21] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge.