Stability of infinite clusters in supercritical percolation |
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Authors: | Roberto H. Schonmann |
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Affiliation: | (1) Mathematics Department, University of California at Los Angeles, Los Angeles, CA 90095, USA (e-mail: rhs@math.ucla.edu), US |
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Abstract: | . A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p 1<p 2≤1 and percolation occurs at level p 1, then every infinite cluster at level p 2 contains some infinite cluster at level p 1. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p 1 there is a unique infinite cluster then the same holds at level p 2. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls. |
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Keywords: | Mathematics Subject Classification (1991): Primary 60K35 |
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