Random motion in random or irregular media

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The aim of this work is to study the behaviour of a randomly moving particle in an irregular medium. This can be regarded as a simple model for diffusion of particulates in rock or soil. The mathematically rigorous analysis of such models has proved difficult and there has been a lot recent activity as new techniques have been developed and applied. The aim is to develop an understanding of how the irregularities in the medium affect the properties of the diffusion and in particular to estimate such quantities as the probability that a particle started at some point has reached another at a certain time. The percolation model is a simple mathematical model for a random medium. For each edge of the square lattice we toss a coin and declare the edge to be open with a probability p, otherwise it is closed. As p increases the size of the open clusters increases until at a specific value an infinite cluster will form. This infinite cluster has fractal structure which makes diffusion in this cluster rather different from that in normal space and we hope to gain a further understanding of this diffusion in this and related models.