 98705 Roberto H. Schonmann
 Stability of infinite clusters in supercritical percolation
(169K, postscript)
Nov 10, 98

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Abstract. A recent theorem by H\"aggstr\"om and Peres concerning independent
percolation is extended to all the quasitransitive 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.
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