Biased random walk among random conductances

We consider the effective diffusivity (i.e. the covariance matrix in the central limit theorem) of a random walk among random conductances. It is interesting and non-trivial to describe this diffusivity in terms of the law of the conductances. The Einstein relation gives a different interpretation of the effective diffusivity as mobility. The mobility measures the response of the diffusing particle to a constant exterior force: Consider the perturbed process obtained by imposing a constant drift of strength $\lambda$ in some fixed direction. The perturbed motion satisfies (as one can show in many examples) a law of large numbers with effective velocity $v(\lambda)$. The mobility is the derivative of $v(\lambda)$ as $\lambda$ goes to 0. The Einstein relation says that the mobility and the diffusivity of a particle coincide. The Einstein relation is conjectured to hold for a variety of models, but it is proved insofar only for particular cases. We explain how it follows from an expansion of the invariant measures for the environment, seen from the particle.

The talk is based on joint work with Jan Nagel and Xiaoqin Guo.