Random walks in irregular media

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The study of random walks in mathematics goes back as least as far as the work of Polya in 1921; he showed that a simple random walk in one or two dimensions will return to its starting point infinitely often, while in 3 or more dimensions it will ultimately leave its initial neighbourhood, never to return. At large length scales random walks can be approximated by Brownian motion. The connection between diffusion and partial differential equations (PDE) is due to Einstein; the transition probabilities of the diffusion process are solutions to the heat equation. These connections, which hold in great generality, were explored very thoroughly in the second part of the 20th century by mathematicians. The proposed research is to study random walks (or other diffusion processes) in irregular media. An example would be an infinite road grid, where according to some stochastic mechanism a proportion of the streets are blocked. (In the case when this is completely at random this is the percolation model.) Two main types of behaviour can arise. The first is that the obstacles (i.e. irregularities) just have a local effect, so that viewed from a great distance (i.e. at large length scales) they play no essential role, and the random walker just appears as a particle moving in a homogeneous medium. The second possibility is that the effect of the irregularity persists at all length scales. This often occurs when the limiting object has fractal properties. In these circumstances the irregularity of the medium acts to slow down the particle; its displacement grows more slowly than that of the simple random walk. This phenomena is often called 'anomalous diffusion'. The proposed research will consider both types of behaviour. In the first case, the problems are to study general kinds of irregular environments and (as far as it remains true) to prove two types of regularity for the large scale (or more precisely limiting) process. The first kind of regularity is an extension of the classical Central Limit Theorem - the particle displacement at large times should be close to a Brownian motion. The second kind of regularity is that the transition densities of the particle motion can be controlled by multiples of the Gaussian distribution. If both kinds of regularity occur then one has an very good description of the particle motion. In the second case, the overall aim is to find methods for computing the behaviour of the random process given knowledge of the kind of graph irregularity. For exampe, one would like to compute the 'walk dimension', which describes the rate at which (on average) a particle moves from its starting point. One interesting set of examples are uniform spanning trees. These can be tackled theoretically because the branches of the trees can be described by random walk paths with the loops erased.