Bond percolation processes

Abstract

Bernoulli percolation processes model flow in a randomly heterogeneous medium by opening or closing connections, called bonds, between sites of a d-dimensional lattice. Flow is then allowed to occur instantaneously along open bonds, and one is interested in the extent of the volumes reached by the fluid, these volumes being known as clusters. A fundamental result of the theory is that there exists a critical value, related to the probability of bond occupation, which separates, in the parameter space, regions of local and global spread of the fluid. The thesis presents the definitions and certain of the methodology of bond percolation theory, and then reviews rigorous results, with particular emphasis on models which yield results of the greatest generality. Differential inequalities for various quantities are considered, and used to prove that the critical value described above is unique. Also it is known that, above criticality, should a cluster of infinite extent occur, then, with probability one, there is only one such cluster. An important result, particularly for applications, is the exponential decay, below criticality, of the cluster size distribution. Finally, using a more restrictive model, results are obtained concerning the surface to volume ratio of infinite clusters, and the sub-exponential decay of the cluster size distribution above criticality. Invasion percolation, a stochastic growth model, is the most recently introduced form of percolation theory. In this model, flow occurs one bond at a time, each bond being assigned a resistance, and the fluid taking the path of least resistance. The theory of the nearest neighbour case is considered, and results from Bernoulli percolation used to derive bounds on, and values for, various time dependent quantities. An extension of the theory to a more general model is briefly discussed.