Abstract

The random conductance model is a well-established model for a random walk in random environment. During the last decade the question whether a quenched invariance principle (or quenched functional central limit theorem) holds for the random walk and whether the associated heat kernel satisfies a quenched local limit theorem has been intensively studied. In situations where the environment is generated by unbounded conductances these questions turned out to be rather non-trivial because of the possibility that the random walk might get trapped. In this talk we will review recent results for the case of an ergodic, degenerate environment. We present a quenched invariance principle and a quenched local limit theorem for ergodic conductances satisfying a certain moment condition. This talk is based on joint work with Jean-Dominique Deuschel and Martin Slowik.