The Random Conductance Model

In the random conductance model, each edge of the hypercubic lattice is assigned a positive and finite random variable called the conductance. The resulting (random) resistor network is directly linked to a Markov chain --- referred to, invariably, as a random walk among random conductances --- where at each step the walk chooses a neighbor at random with probability proportional to the conductance of the corresponding edge. The conductances are distributed according to a translation-invariant, or even iid, law; the properties of the random walk are studied against a typical sample from this law (these are the so called quenched problems).

In my three lectures I will discuss the following facts/situations:
(1) The proof of recurrence and transience for this random walk and the connection with effective resistivity.
(2) The proof of an invariance principle for the path of the random walk in the case when the conductances are bounded away from zero and infinity (so called elliptic case).
(3) Extensions to non-elliptic situations (random walk on the supercritical percolation cluster, regular and anomalous heat-kernel decay).

I will finish with an outlook of problems awaiting solutions and possible directions of future research. My contributions to this field are based on joint works with N. Berger, T. Prescott, G. Kozma and C. Hoffman.