×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI arXiv
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.