Stability of infinite clusters in supercritical percolation

. 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.