Angel, Omer; Goodman, Jesse; Merle, Mathieu Scaling limit of the invasion percolation cluster on a regular tree. (English) Zbl 1281.60076 Ann. Probab. 41, No. 1, 229-261 (2013). The authors study the invasion percolation cluster as a metric space with respect to graph distance. They prove that the rescale rooted incipient infinite cluster has a scaling limit w.r.t. the pointed Gramov-Hausdorff topology. The limit is a random real tree and its contour and height functions are described as certain diffusive processes. Using this convergence, the authors make precise certain asymptotic results for the invasion percolation cluster. Reviewer: Anatoliy Pogorui (Zhytomyr) Cited in 8 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation Keywords:percolation; cluster; Gramov-Hausdorff topology; diffusive process × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Aldous, D. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1 228-266. · Zbl 0733.60016 · doi:10.1214/aoap/1177005936 [2] Angel, O., Goodman, J., den Hollander, F. and Slade, G. (2008). Invasion percolation on regular trees. Ann. Probab. 36 420-466. · Zbl 1145.60050 · doi:10.1214/07-AOP346 [3] Barlow, M. T. and Kumagai, T. (2006). Random walk on the incipient infinite cluster on trees. Illinois J. Math. 50 33-65 (electronic). · Zbl 1110.60090 [4] Blumenthal, R. M. (1992). Excursions of Markov Processes . Birkhäuser, Boston, MA. · Zbl 0983.60504 [5] Chayes, J. T., Chayes, L. and Newman, C. M. (1985). The stochastic geometry of invasion percolation. Comm. Math. Phys. 101 383-407. · Zbl 0596.60096 · doi:10.1007/BF01216096 [6] Damron, M. and Sapozhnikov, A. (2011). Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theory Related Fields 150 257-294. · Zbl 1225.82030 · doi:10.1007/s00440-010-0274-y [7] Duquesne, T. (2005). Continuum tree limit for the range of random walks on regular trees. Ann. Probab. 33 2212-2254. · Zbl 1099.60021 · doi:10.1214/009117905000000468 [8] Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281 vi+147. · Zbl 1037.60074 [9] Goodman, J. (2012). Exponential growth of ponds in invasion percolation on regular trees. J. Stat. Phys. . · Zbl 1246.82047 · doi:10.1007/s10955-012-0509-7 [10] Jacod, J. (1985). Théorèmes limite pour les processus. In École D’été de Probabilités de Saint-Flour , XIII- 1983. Lecture Notes in Math. 1117 298-409. Springer, Berlin. · Zbl 0565.60030 [11] Járai, A. A. (2003). Invasion percolation and the incipient infinite cluster in 2D. Comm. Math. Phys. 236 311-334. · Zbl 1041.82020 · doi:10.1007/s00220-003-0796-6 [12] Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369-394. · Zbl 0584.60098 · doi:10.1007/BF00776239 [13] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245-311. · Zbl 1189.60161 · doi:10.1214/154957805100000140 [14] Munn, M. (2010). Volume growth and the topology of pointed Gromov-Hausdorff limits. Differential Geom. Appl. 28 532-542. · Zbl 1194.53030 · doi:10.1016/j.difgeo.2010.04.004 [15] Newman, C. M. and Stein, D. L. (1995). Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E (3) 51 5228-5238. [16] Nickel, B. and Wilkinson, D. (1983). Invasion percolation on the Cayley tree: Exact solution of a modified percolation model. Phys. Rev. Lett. 51 71-74. · doi:10.1103/PhysRevLett.51.71 [17] Rogers, L. C. G. and Williams, D. (1994). Diffusions , Markov Processes and Martingales , 2nd ed. Cambridge Mathematical Library 2 . Cambridge Univ. Press, Cambridge. · Zbl 0826.60002 [18] Wilkinson, D. and Willemsen, J. F. (1983). Invasion percolation: A new form of percolation theory. J. Phys. A 16 3365-3376. · doi:10.1088/0305-4470/16/14/028 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.