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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Received: 22 December 1997 / Revised version: 9 July 1998
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Schonmann, R. Stability of infinite clusters in supercritical percolation. Probab Theory Relat Fields 113, 287–300 (1999). https://doi.org/10.1007/s004400050209
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DOI: https://doi.org/10.1007/s004400050209
- Mathematics Subject Classification (1991): Primary 60K35
- Keywords: Percolation, quasi-transitive graphs, continuity of percolation probability, monotonicity of uniqueness