Vacant set of random interlacements and percolation. (English) Zbl 1202.60160

The paper deals with the model of random interlacements consisting of a countable collection of doubly infinite trajectories on a multidimensional integer lattice. A nonnegative parameter \(u\) measures how many trajectories enter the picture. The union of the supports of these trajectories defines the interlacement at level \(u\). It is an infinite connected translation invariant random subset of the lattice. The author introduces a critical value \(u^{\ast}\) such that the vacant set percolates for \(u\) less then \(u^{\ast}\) and does not percolate for \(u\) greater then \(u^{\ast}\). The main results of the paper show that \(u^{\ast}\) is finite for three-dimensional lattice and strictly positive when the lattice dimension is greater than 6.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
Full Text: DOI arXiv Link


[1] D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs
[2] I. Benjamini and A. Sznitman, ”Giant component and vacant set for random walk on a discrete torus,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 10, iss. 1, pp. 133-172, 2008. · Zbl 1141.60057
[3] E. Bolthausen and J. Deuschel, ”Critical large deviations for Gaussian fields in the phase transition regime. I,” Ann. Probab., vol. 21, iss. 4, pp. 1876-1920, 1993. · Zbl 0801.60018
[4] M. J. A. M. Brummelhuis and H. J. Hilhorst, ”Covering of a finite lattice by a random walk,” Phys. A, vol. 176, iss. 3, pp. 387-408, 1991.
[5] R. M. Burton and M. Keane, ”Density and uniqueness in percolation,” Comm. Math. Phys., vol. 121, iss. 3, pp. 501-505, 1989. · Zbl 0662.60113
[6] A. Dembo and A. Sznitman, ”On the disconnection of a discrete cylinder by a random walk,” Probab. Theory Related Fields, vol. 136, iss. 2, pp. 321-340, 2006. · Zbl 1105.60029
[7] A. Dembo and A. Sznitman, ”A lower bound on the disconnection time of a discrete cylinder,” in In and Out of Equilibrium. 2, Basel: Birkhäuser, 2008, pp. 211-227. · Zbl 1173.82360
[8] G. Grimmett, Percolation, Second ed., New York: Springer-Verlag, 1999. · Zbl 0926.60004
[9] O. Häggström and J. Jonasson, ”Uniqueness and non-uniqueness in percolation theory,” Probab. Surv., vol. 3, pp. 289-344, 2006. · Zbl 1189.60175
[10] G. A. Hunt, ”Markoff chains and Martin boundaries,” Illinois J. Math., vol. 4, pp. 313-340, 1960. · Zbl 0094.32103
[11] G. F. Lawler, Intersections of Random Walks, Boston, MA: Birkhäuser, 1991. · Zbl 0925.60078
[12] E. W. Montroll, ”Random walks in multidimensional spaces, especially on periodic lattices,” J. Soc. Indust. Appl. Math., vol. 4, pp. 241-260, 1956. · Zbl 0080.35101
[13] C. M. Newman and L. S. Schulman, ”Infinite clusters in percolation models,” J. Statist. Phys., vol. 26, iss. 3, pp. 613-628, 1981. · Zbl 0509.60095
[14] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, New York: Springer-Verlag, 1987. · Zbl 0633.60001
[15] V. Sidoravicius and A. Sznitman, ”Percolation for the vacant set of random interlacements,” Comm. Pure Appl. Math., vol. 62, iss. 6, pp. 831-858, 2009. · Zbl 1168.60036
[16] M. L. Silverstein, Symmetric Markov Processes, New York: Springer-Verlag, 1974, vol. 426. · Zbl 0296.60038
[17] F. Spitzer, Principles of Random Walks, Second ed., New York: Springer-Verlag, 1976, vol. 34. · Zbl 0359.60003
[18] F. Spitzer, Differential Geometry, Berkeley, CA: Publish or Perish, 1979, vol. 2.
[19] A. Sznitman, ”How universal are asymptotics of disconnection times in discrete cylinders?,” Ann. Probab., vol. 36, iss. 1, pp. 1-53, 2008. · Zbl 1134.60061
[20] A. Sznitman, ”Random walks on discrete cylinders and random interlacements,” Probab. Theory Related Fields, vol. 145, iss. 1-2, pp. 143-174, 2009. · Zbl 1172.60316
[21] A. Sznitman, ”Upper bound on the disconnection time of discrete cylinders and random interlacements,” Ann. Probab., vol. 37, pp. 1715-1746, 2009. · Zbl 1179.60025
[22] A. Teixeira, ”On the uniqueness of the infinite cluster of the vacant set of random interlacements,” Ann. Appl. Probab., vol. 19, iss. 1, pp. 454-466, 2009. · Zbl 1158.60046
[23] A. Telcs, The Art of Random Walks, New York: Springer-Verlag, 2006. · Zbl 1104.60003
[24] M. Weil, ”Quasi-processus,” in Séminaire de Probabilités, IV (Univ. Strasbourg, 1968/69), New York: Springer-Verlag, 1970, vol. 124, pp. 216-239. · Zbl 0211.21203
[25] D. Windisch, ”Random walk on a discrete torus and random interlacements,” Electron. Commun. Probab., vol. 13, pp. 140-150, 2008. · Zbl 1187.60089
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.