98-703 Roberto H. Schonmann

Percolation in $\infty + 1$ dimensions at the uniqueness threshold (166K, postcript) Nov 10, 98

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Abstract. For independent density $p$ site percolation on the (transitive non-amenable) graph $\T_b \times \Z$, where $\T_b$ is a homogeneous tree of degree $b+1$, and $b$ is supposed to be large, it is shown that for $p= p_u = \inf \{p \colon \text{a.s. there is a unique infinite cluster}\}$ there are a.s. infinitely many infinite clusters. This contrasts with a recent result of Benjamini and Schramm, according to whom for transitive non-amenable planar graphs there is a.s. a unique infinite cluster at $p_u$.

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