Three properties of the infinite cluster in percolation theory

Abstract

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 ac. A limiting form for the multiplicity of finite clusters for aa_c, dS(a)/dac) is the maximum value of S(a) and that its first derivative is discontinuous at a=a_c.