In the first talk we aim to provide a gentle introduction to 2-scale convergence for random walks in a random environment with symmetric rates. Using the Palm theory of random measures we discuss ergodicity issues at a 2-scale level. We then introduce a basic difference calculus and define 2-scale convergence of functions and gradients, corresponding to an enforced averaging property. Finally we discuss the fundamental compactness and structure theorem for bounded families of H^1-functions, where geometrical issues of square integrable forms and the homogenized matrix emerge.

In the second talk we discuss some applications of the above 2-scale convergence for random walks in a random environment. A first one is given by the invariance principle of random walks on the supercritical percolation cluster of P. Mathieu and A. Piatnitski. We then discuss more in detail applications to the homogenization of the massive Poisson equation associated with the random walk, which also enters in the derivation of the hydrodynamic limit of exclusion and zero range processes. Finally, we discuss applications to random resistor networks, in particular to the conductance model and the Miller-Abrahams one associated to Mott variable range hopping in amorphous solids.