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Cylinders’ percolation in three dimensions. (English) Zbl 1333.60205
Summary: We study the complementary set of a Poissonian ensemble of infinite cylinders in \({\mathbb {R}}^3\), for which an intensity parameter \(u > 0\) controls the amount of cylinders to be removed from the ambient space. We establish a non-trivial phase transition, for the existence of an unbounded connected component of this set, as \(u\) crosses a critical non-degenerate intensity \(u_*\). We moreover show that this complementary set percolates in a sufficiently thick slab, in spite of the fact that it does not percolate in any given plane of \({\mathbb {R}}^3\), regardless of the choice of \(u\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
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[1] Broman, E.I., Tykesson, J.: Connectedness of poisson cylinders in euclidean space (2013 preprint). arXiv:1304.6357 · Zbl 1333.60197
[2] Burton, R.M., Keane, M.: Density and uniqueness in percolation. Commun. Math. Phys. 121(3), 501-505 (1989) · Zbl 0662.60113
[3] Gács, P, The clairvoyant demon has a hard task, Comb. Probab. Comput., 9, 421-424, (2000) · Zbl 0974.60091
[4] Grimmett, G.: Percolation. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 321, 2nd edn. Springer, Berlin (1999) · Zbl 1263.82027
[5] Hall, P, On continuum percolation, Ann. Probab., 13, 1250-1266, (1985) · Zbl 0588.60096
[6] Hilário, M.R.: Coordinate percolation on \(\mathbb{Z}^3\). PhD thesis, IMPA (2011) · Zbl 1231.60117
[7] Lawler, GF; Trujillo Ferreras, JA, Random walk loop soup, Trans. Am. Math. Soc., 359, 767-787, (2007) · Zbl 1120.60037
[8] Lawler, G.F., Werner, Wendelin: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565-588 (2004) · Zbl 1049.60072
[9] Le Jan, Y.: Markov Paths, Loops and Fields. Lecture Notes in Mathematics, vol. 2026. Springer, Heidelberg (2011). Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. (Saint-Flour Probability Summer School) · Zbl 1202.60160
[10] Lyons, R; Schramm, O, Indistinguishability of percolation clusters, Ann. Probab., 27, 1809-1836, (1999) · Zbl 0960.60013
[11] Nacu, Ş; Werner, W, Random soups, carpets and fractal dimensions, J. Lond. Math. Soc. (2), 83, 789-809, (2011) · Zbl 1223.28012
[12] Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. Springer, New York (2008). (Reprint of the 1987 original)
[13] Rolla, L.T., Werner, W.: Private communication (2011) · Zbl 0960.60013
[14] Sidoravicius, V; Sznitman, A-S, Percolation for the vacant set of random interlacements, Commun. Pure Appl. Math., 62, 831-858, (2009) · Zbl 1168.60036
[15] Sidoravicius, V; Sznitman, A-S, Connectivity bounds for the vacant set of random interlacements, Ann. Inst. Henri Poincaré Probab. Stat., 46, 976-990, (2010) · Zbl 1210.60107
[16] Symanzik, K; Jost, R (ed.), Euclidean quantum feild theory, (1969), New York
[17] Szász, D. (ed.): Hard Ball Systems and the Lorentz Gas. Encyclopaedia of Mathematical Sciences, vol. 101. Mathematical Physics, II. Springer, Berlin (2000) · Zbl 0953.00014
[18] Sznitman, A-S, Vacant set of random interlacements and percolation, Ann. Math. (2), 171, 2039-2087, (2010) · Zbl 1202.60160
[19] Teixeira, A, On the uniqueness of the infinite cluster of the vacant set of random interlacements, Ann. Appl. Probab., 19, 454-466, (2009) · Zbl 1158.60046
[20] Teixeira, A, On the size of a finite vacant cluster of random interlacements with small intensity, Probab. Theory Relat. Fields, 150, 529-574, (2011) · Zbl 1231.60117
[21] Tykesson, J; Windisch, D, Percolation in the vacant set of Poisson cylinders, Probab. Theory Relat. Fields, 154, 165-191, (2012) · Zbl 1263.82027
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