Percolation in ∞+1 Dimensions at the Uniqueness Threshold

: For independent density p site percolation on the (transitive non-amenable) graph T b \Theta 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 = inffp: a.s. there is a unique infinite clusterg 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 . 1. Introduction and results Percolation theory has for a long time been a central area of research in both Probability and Mathematical Physics. Two complementary reasons concur for this. In the words of Kesten in the preface to [Kes2]: " ... it is a source of fascinating problems of the best kind a mathematician can wish for: problems which are easy to state with a minimum of preparation, but whose solutions are (apparently) difficult and require new methods. At the same time many of the problems are of interest t..