Journal of Statistical Physics

, Volume 130, Issue 5, pp 983–1009 | Cite as

Sharpness of the Phase Transition and Exponential Decay of the Subcritical Cluster Size for Percolation on Quasi-Transitive Graphs



We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the Aizenman & Barsky method.


Random graphs Edge percolation Site percolation Quasi-transitive graphs Phase transition 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Emmy-Noether-Programme of the Deutsche Forschungsgemeinschaft & Fakultät für Mathematik, TU ChemnitzChemnitzGermany

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