Random walks in irregular media

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My proposed research is on random walks in random environments, particularly on environments which arise in statistical physics. I will study the random conductance model, a model for a random walk in a random environment. The environment is made by giving the edges in Z^d non-negative random weights. The walk then jumps across an edge at a rate proportional to the edge weight. The main problems are to prove that the rescaled random walk converges to a multiple of Brownian motion, and to obtain Gaussian bounds on the transition density of the walk. If both properties hold, then combining them gives very good information concerning the process - one obtains Harnack inequalities, a local limit theorem, and good bounds on Green's functions. The case of independent weights has recently been solved, but many hard problems remain if one considers stationary ergodic edge weights, or edge weights which vary in time. A second question is to consider analogous problems in the continuum context - a particular problem here is the motion of a Brownian particle with reflection from a random collection of obstacles. The final topic is to study random walks on various random graphs, at or near criticality. Unlike the cases described above, one does not expect a Brownian limit, since these graphs have fractal properties. One particular problem is the uniform spanning tree in high dimensions.