Roberto H. Schonmann
Stability of infinite clusters in supercritical percolation
(169K, postscript)

ABSTRACT.  A recent theorem by H\"aggstr\"om and Peres concerning independent 
percolation is extended to all the quasi-transitive graphs. 
This theorem states that if $0 < p_1 < p_2 \leq 1$ 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.