Three properties of the infinite cluster in percolation theory

The author presents proofs of the following properties of the infinite cluster in percolation theory for all p>p c and for all lattices: (i) the ratio of the number of boundary sites to cluster sites is (1-p)/p; (ii) the specific logarithmic multiplicity of the infinite cluster per cluster site is equal to the specific logarithmic multiplicity of all configurations of the lattice per occupied site, 1np+((1-p)/p)ln(1-p); and (iii) the specific logarithmic multiplicity of the infinite cluster per cluster site is given by S(a)=(1+a)ln(1+a)-alna for all aa c , dS(a)/da