Group-invariant percolation on graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fu ndamental tool is a new mass-transport technique that has been occasionally used elsewhere and is developed further here. Perhaps surprisingly, these investigations of group-invariant percolation p roduce results that are new in the Bernoulli setting. Most notably, we prov e that critical Bernoulli percolation on any nonamenable Cayley graph has n o infinite clusters. More generally, the same is true for any nonamenable g raph with a unimodular transitive automorphism group. We show that G is amenable iff for all alpha < 1, there is a G-invariant si te percolation process w on X with P[x is an element of w] > alpha for all vertices re and with no infinite components. When G is not amenable, a thre shold a < 1 appears. An inequality for the threshold in terms of the isoper imetric constant is obtained, extending an inequality of Haggstrom for regu lar trees. If G acts transitively on X, we show that G is unimodular iff the expected degree is at least 2 in any G-invariant bond percolation on X with all comp onents infinite. The investigation of dependent percolation also yields some results on auto morphism groups of graphs that do not involve percolation.