  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.
