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Density and uniqueness in percolation. (English) Zbl 0662.60113
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 C. M. Newman and 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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##### References:
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