Percolation on nonamenable products at the uniqueness threshold (Q1577466)

Let \(X\) and \(Y\) be infinite quasi-transitive graphs such that the automorphism group of \(X\) is not amenable. For i.i.d. percolation on the direct product \(X\times Y\), the author shows that the set of retention parameters \(p\), where a.s. there is a unique infinite cluster, does not contain its infimum \(p_u\). This extends a result of Schonmann, who considered the direct product of a regular tree and \(Z\).