Roberto H. Schonmann
Percolation in $\infty + 1$ dimensions at the uniqueness threshold
(166K, postcript)
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 $p_u$.