In this paper and its sequel we consider the relation between various conductivity problems and their associated percolation models. The emphasis of this paper is on the methods of solving percolation problems. Toward that end we define the percolation problem and propose a classification for several types of models. This classification scheme is useful for regular and random lattices and shows the connection between bond and site percolation models. With the aid of a large computer we have used Monte Carlo techniques to solve many random-lattice percolation models in two and three dimensions. These models illustrate the effects of hard-core interactions between sites, of deterministic and probabilistic bonding parameters, and of changing the functional form of the bonding criterion. The random-lattice solutions are used to examine the critical volume fraction concept and the critical number of bonds per site ($B_c$) concept. These concepts have been previously proposed as methods of obtaining solutions to percolation problems. We find that, subject to several conditions, these concepts may be used for approximate solutions.

• Received 7 September 1973

DOI:https://doi.org/10.1103/PhysRevB.10.1421