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

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Abstract

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.

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Author notes

  1. Tonći Antunović

    Present address: Department of Mathematics, UC Berkeley, Berkeley, CA, 94720, USA

  2. Ivan Veselić

    Present address: Institut für Angewandte Mathematik, Universität Bonn, 53115, Bonn, Germany

Authors and Affiliations

  1. Emmy-Noether-Programme of the Deutsche Forschungsgemeinschaft & Fakultät für Mathematik, TU Chemnitz, 09107, Chemnitz, Germany

    Tonći Antunović & Ivan Veselić

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  1. Tonći Antunović
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  2. Ivan Veselić
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Correspondence to Tonći Antunović.

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Antunović, T., Veselić, I. Sharpness of the Phase Transition and Exponential Decay of the Subcritical Cluster Size for Percolation on Quasi-Transitive Graphs. J Stat Phys 130, 983–1009 (2008). https://doi.org/10.1007/s10955-007-9459-x

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