98-705 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 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.

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