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Random walks on discrete cylinders with large bases and random interlacements. (English) Zbl 1191.60062
A.-S. Sznitman [Probab. Theory Relat. Fields 145, No. 1–2, 143–174 (2009; Zbl 1172.60316)] has introduced the model of random interlacements on $$\mathbb{Z}^{d+1}, d>2$$, and explored its relation with the microscopic structure left by simple random walk on an infinite discrete cylinder $$(\mathbb{Z}/N\mathbb{Z})^d \times Z$$ by times of order $$N^{2d}$$ . That is generalised to random walk trajectories on a cylinder of the form $$G_N\times \mathbb{Z}$$, where $$G_N$$ is a large finite connected weighted graph, running to a time of order $$|G_N|^2$$, Under suitable assumptions , the set of points not visited by the random walk, in a neighborhood od a point with the $$\mathbb{Z}$$ component of order $$|G_N|$$, converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the local structure of the graph. As examples of $$G_N$$, Sierpinski gasket and $$d$$-ary tree are considered.

##### MSC:
 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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##### References:
 [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. · Zbl 1083.60060 [3] Chung, K. L. (1974). A Course in Probability Theory , 2nd ed. Academic Press, New York. · Zbl 0345.60003 [4] Darling, R. W. R. and Norris, J. R. (2008). Differential equation approximations for Markov chains. Probab. Surv. 5 37-79. · Zbl 1189.60152 [5] Durrett, R. (2005). Probability : Theory and Examples , 3rd ed. Brooks/Cole, Belmont. · Zbl 1202.60002 [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. · Zbl 0326.60065 [7] Jones, O. D. (1996). Transition probabilities for the simple random walk on the Sierpiński graph. Stochastic Process. Appl. 61 45-69. · Zbl 0853.60058 [8] Khaśminskii, R. Z. (1959). On positive solutions of the equation U + Vu =0. Theory Probab. Appl. 4 309-318. · Zbl 0089.34501 [9] Lawler, G. F. (1991). Intersections of Random Walks . Birkhäuser, Boston, MA. · Zbl 1228.60004 [10] Norris, J. R. (1997). Markov Chains . Cambridge Univ. Press, New York. · Zbl 0873.60043 [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. · Zbl 0456.60029 [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. · Zbl 0885.60061 [13] Shima, T. (1991). On eigenvalue problems for the random walks on the Sierpiński pre-gaskets. Japan J. Indust. Appl. Math. 8 127-141. · Zbl 0715.60088 [14] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831-858. · Zbl 1168.60036 [15] Sznitman, A.-S. (1999). Slowdown and neutral pockets for a random walk in random environment. Probab. Theory Related Fields 115 287-323. · Zbl 0947.60095 [16] Sznitman, A. S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2). To appear. Available at http://www.math.ethz.ch/u/sznitman/preprints. · Zbl 1202.60160 [17] Sznitman, A.-S. (2009). Random walks on discrete cylinders and random interlacements. Probab. Theory Related Fields 145 143-174. · Zbl 1172.60316 [18] Sznitman, A. S. (2009). Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37 1715-1746. · Zbl 1179.60025 [19] Teixeira, A. (2009). Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 no. 54, 1604-1628. · Zbl 1192.60108 [20] Telcs, A. (2006). The Art of Random Walks. Lecture Notes in Math. 1885 . Springer, Berlin. · Zbl 1104.60003 [21] Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138 . Cambridge Univ. Press, Cambridge. · Zbl 0951.60002
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