Random walks, the heat equation, and percolation.

Abstract: I will discuss `homogenization' of the heat equation in a random environment. The environment is given by bond percolation on the Euclidean lattice Z^d: each bond is present(or open) with probability p, and closed (absent) with probability 1-p, independently of all other bonds. When p is large enough, this graph has a single infinite connected component, but this has arbitrarily large holes and bottlenecks. Nevertheless we can now say a great deal about the heat equation (or the random walk) on this set. The talk will end with some simulations.