Density and uniqueness in percolation

Abstract

Two results on site percolation on thed-dimensional lattice,d≧1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster has a well-defined density with probability one. The second theorem states that if in addition, the probability measure satisfies the finite energy condition of Newman and Schulman, then there can be at most one infinite cluster with probability one. The simple arguments extend to a broad class of finite-dimensional models, including bond percolation and regular lattices.