We consider a random walk on the d-dimensional lattice in the random conductance model. Each edge of the lattice is assigned randomly a conductance and for a fixed realization of this environment, the random walker crosses an edge with a probability proportional to the conductivity of the edge. This model is one of the prime examples of a reversible process in an inhomogeneous medium. When we introduce a bias to the right, the process satisfies a law of large numbers with a nonzero effective speed. We are interested in properties of the speed as a function of the bias. For example, is the speed continuous, and is it increasing in the strength of the bias?
We will discuss general ideas how to deal with such a random medium and how it leads to some atypical behavior. No prior knowledge of random walks in random environments is assumed. The talk is based on joint works with Noam Berger, Nina Gantert and Xiaoqin Guo.