Critical Probabilities for Cluster Size and Percolation Problems

When particles occupy the sites or bonds of a lattice at random with probability p, there is a critical probability pc above which an infinite connected cluster of particles forms. Rigorous bounds and inequalities are obtained for pc on a variety of lattices and compared briefly with previous numerical estimates. In particular, by extending Harris' work, it is proved that pc(s,L2)⩾12 for the site problem on a plane lattice L2 (without crossing bonds), while for the bond problem pc(b,L2)+pc(b,L2D)⩾1 where L2D is the dual lattice to L2. Simple arguments demonstrate that the bond problem is a special case of the site problem and that the critical probabilities for the bond problem on the plane square and triangular lattice cannot exceed those for the corresponding site problems.