Examples of invariant processes on Cayley graphs

An invariant process on a Cayley graph G is a random labeling of its vertices which is invariant under group multiplications. In this work we answer three natural questions about invariant processes on Cayley graphs. The first asks if there is an invariant planar percolation where both the open and closed vertices span a connected subgraph with critical probability strictly less than 1. The second asks if the existence of a monotone coupling between two invariant random subgraphs implies the existence of a monotone coupling which is also invariant? The third one asks if a factor of independent uniform labeling which also has uniform margins then the components spanned by identically labeled vertices must be finite? We show that in each case a relevant example does exist. In the way the questions are formulated this means that the answer to the first question is "yes" while the answer for the second and third is "no".