Density and uniqueness in percolation (Q1113557)

Two results on site percolation on the d-dimensional lattice, \(d\geq 1\) 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 \textit{C. M. Newman} and \textit{L. S. Schulman} [J. Stat. Phys. 26, 613-628 (1981; Zbl 0509.60095)], 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.