A conditionally sure ergodic theorem with an application to percolation |
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Authors: | Michael Keane Masato Takei |
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Affiliation: | 1. Department of Mathematics, Wesleyan University, Middletown, CT 06459-0128, United States;2. Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands;3. Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, 79-5, Tokiwa-Dai, Hodogaya, Yokohama 240-8501, Japan |
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Abstract: | We prove an apparently new type of ergodic theorem, and apply it to the site percolation problem on sparse random sublattices of Zd (d≥2), called “lattices with large holes”. We show that for every such lattice the critical probability lies strictly between zero and one, and the number of the infinite clusters is at most two with probability one. Moreover for almost every such lattice, the infinite cluster, if it exists, is unique with probability one. |
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Keywords: | Ergodic theorem Percolation Fractal-like graphs |
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