Critical Probabilities for Cluster Size and Percolation Problems

Abstract

When particles occupy the sites or bonds of a lattice at random with probability p, there is a critical probability p_c above which an infinite connected cluster of particles forms. Rigorous bounds and inequalities are obtained for p_c on a variety of lattices and compared briefly with previous numerical estimates. In particular, by extending Harris' work, it is proved that $p_c{\mathrm{(s,L}}_2\mathrm{)⩾}\frac{1}{2}$ for the site problem on a plane lattice L₂ (without crossing bonds), while for the bond problem $p_c{\mathrm{(b,L}}_2{\mathrm{)+p}}_c{\mathrm{(b,L}}_2^D\text{)⩾1}$ where $L_2^D$ 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.