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Stability of infinite clusters in supercritical percolation
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  • Published: February 1999

Stability of infinite clusters in supercritical percolation

  • Roberto H. Schonmann1 

Probability Theory and Related Fields volume 113, pages 287–300 (1999)Cite this article

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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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Authors and Affiliations

  1. Mathematics Department, University of California at Los Angeles, Los Angeles, CA 90095, USA (e-mail: rhs@math.ucla.edu), , , , , , US

    Roberto H. Schonmann

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  1. Roberto H. Schonmann
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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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  • Issue Date: February 1999

  • DOI: https://doi.org/10.1007/s004400050209

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  • Mathematics Subject Classification (1991): Primary 60K35
  • Keywords: Percolation, quasi-transitive graphs, continuity of percolation probability, monotonicity of uniqueness
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