The ant in the labyrinth: diffusion in random networks near the percolation threshold

Abstract

The parameters which describe the mean-squared displacement (R²(t)) of a random walker on a random network have a characteristic singular dependence on epsilon =(p-p_c)/p_c near the percolation threshold. The critical exponents, which characterise the singularities of the diffusion constant, moment of inertia of finite clusters, and time constants for development of the long-time behaviour, are related by a scaling theory. They may also be related to the exponent theories for the percolation and percolation conduction problems. An equivalent resistor network can be described which is equivalent to the time Laplace transform of the diffusion problem. These problems will be given explicit treatment for the Cayley tree.