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Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs. (English) Zbl 1214.82028
Summary: 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 and 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 and 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 Aizenman and Barsky’s method.

MSC:
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B43 Percolation
82B27 Critical phenomena in equilibrium statistical mechanics
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