×

zbMATH — the first resource for mathematics

A Harris-Kesten theorem for confetti percolation. (English) Zbl 1323.60130
Summary: Percolation properties of the dead leaves model, also known as confetti percolation, are considered. More precisely, we prove that the critical probability for confetti percolation with square-shaped leaves is \(1/2\). This result is related to a question of I. Benjamini and O. Schramm [Probab. Theory Relat. Fields 111, No. 4, 551–564 (1998; Zbl 0910.60076)] concerning disk-shaped leaves and can be seen as a variant of the Harris-Kesten theorem for bond percolation. The proof is based on techniques developed by B. Bollobás and O. Riordan [Probab. Theory Relat. Fields 136, No. 3, 417–468 (2006; Zbl 1100.60054); Probab. Theory Relat. Fields 140, No. 3–4, 319–343 (2008; Zbl 1135.60057)] to determine the critical probability for Voronoi and Johnson-Mehl percolation.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bálint, Sharp phase transition and critical behaviour in 2D, divide and colour models Stoch Process Appl Vol. 119 pp 937– (2009) · Zbl 1159.60350 · doi:10.1016/j.spa.2008.04.003
[2] Bálint, The high temperature Ising model on the triangular lattice is a critical Bernoulli percolation model, J Stat Phys 139 pp 122– (2010) · Zbl 1189.82019 · doi:10.1007/s10955-010-9930-y
[3] Benjamini, Exceptional planes of percolation, Probab Theory Relat Fields 111 pp 551– (1998) · Zbl 0910.60076 · doi:10.1007/s004400050177
[4] Bollobás, The critical probability for random Voronoi percolation in the plane is 1/2, Probab Theory Related Fields 136 pp 417– (2006) · Zbl 1100.60054 · doi:10.1007/s00440-005-0490-z
[5] Bollobás, Percolation (2006) · Zbl 1118.60001 · doi:10.1017/CBO9781139167383
[6] Bollobás, Sharp thresholds and percolation in the plane, Random Struct Algor 29 pp 524– (2006) · Zbl 1106.60079 · doi:10.1002/rsa.20134
[7] Bollobás, Percolation on random Johnson-Mehl tessellations and related models, Probab Theory Relat Fields 140 pp 319– (2008) · Zbl 1135.60057 · doi:10.1007/s00440-007-0066-1
[8] Bollobás, Erratum to: Percolation on random Johnson-Mehl tessellations and related models, Probab Theory Relat Fields 146 pp 567– (2010) · Zbl 1185.60105 · doi:10.1007/s00440-009-0247-1
[9] Bordenave, The dead leaves model: a general tessellation modeling occlusion, Adv Appl Probab 38 pp 31– (2006) · Zbl 1095.60004 · doi:10.1239/aap/1143936138
[10] Friedgut, Every monotone graph property has a sharp threshold, Proc Am Math Soc 124 pp 2993– (1996) · Zbl 0864.05078 · doi:10.1090/S0002-9939-96-03732-X
[11] Gandolfi, On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation, Ann Probab 16 pp 1147– (1988) · Zbl 0658.60133 · doi:10.1214/aop/1176991681
[12] Grimmett, Percolation, 2. ed. (1999) · Zbl 0926.60004 · doi:10.1007/978-3-662-03981-6
[13] Harris, Coin Tossing, Random Mirrors and Dependent Percolation: Three Paradigms of Phase Transition. PhD thesis (1997)
[14] Hatcher, Algebraic Topology (2002)
[15] C. Hirsch A Harris-Kesten theorem for confetti percolation · Zbl 1323.60130
[16] Jeulin, Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets pp 137– (1997) · Zbl 0944.60010
[17] Last, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probab Theory Relat Fields 150 pp 663– (2011) · Zbl 1233.60026 · doi:10.1007/s00440-010-0288-5
[18] Penrose, Random Geometric Graphs (2003) · Zbl 1029.60007 · doi:10.1093/acprof:oso/9780198506263.001.0001
[19] Roy, Percolation of Poisson sticks on the plane, Probab Theory Relat Fields 89 pp 503– (1991) · Zbl 0725.60115 · doi:10.1007/BF01199791
[20] Svindland, Continuity properties of law-invariant (quasi-)convex risk functions on L, Math Fin Economics 3 pp 39– (2010) · Zbl 1255.91186 · doi:10.1007/s11579-010-0026-x
[21] Berg, Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation, Ann Probab 36 pp 1880– (2008) · Zbl 1155.60044 · doi:10.1214/07-AOP380
[22] Berg, Sharpness of the percolation transition in the two-dimensional contact process, Ann Appl Probab 21 pp 374– (2011) · Zbl 1247.60136 · doi:10.1214/10-AAP702
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.