Mean-field theory and critical exponents for a random resistor network

We consider the conductivities of a random resistor network on a regular lattice near the percolation point. It is shown that there is a close analogy between this problem and that of phase transitions. It is possible to construct a Hamiltonian and define an order parameter. The mean-field equation for the order parameter is found and the conductivities determined in the cases of a network of resistor and open circuits and a network of resistor and shorts. Exponent relations are discussed and the conductivity exponents given in $6-\epsilon$ dimensions. The analog of the resistor network arises in a number of other physical problems, e.g., spin waves in dilute ferromagnets, the elasticity of gels, hopping conductivity, and these results are also of interest in these systems.