Critical percolation on certain nonunimodular graphs

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the amenable cases Z and Z for large d, as well as for all non-amenable graphs with unimodular automorphism groups. We show that the conjecture holds for several classes of non-amenable graphs with non-unimodular automorphism groups: for decorated trees, for the non-unimodular Diestel-Leader graphs, and for direct products of these graphs with an arbitrary transitive graph. We also show that, in any of these graphs, the connection probability between two vertices decay exponentially in their distance. Finally, we prove that critical percolation on the positive part of the lamplighter group has no infinite clusters.