 Percolation uniqueness on an amenable quasi-transitive graph

For a connected infinite locally finite graph that is also an amenable graph and a quasi-transitive graph, an invariant independent site percolation law with every orbit parameter strictly between $0$ and $1$ has either no infinite percolation cluster or exactly one, almost surely. Invariance and independence give a zero–one law. The finite-energy property of Bernoulli percolation excludes a fixed finite number greater than one by opening a finite joining region. Infinitely many clusters would create an encounter box in percolation with positive probability; translated disjoint boxes then have positive density, contradicting the boundary count in a Følner sequence. The finite-energy hypothesis matters if some orbit parameters are $0$ or $1$.