Brownian motion limit of random walks in symmetric non-homogeneous media

The phenomenon of macroscopic homogenization is illustrated with a simple e xample of diffusion. We examine the conditions under which a d-dimensional simple random walk in a symmetric random media converges to a Brownian moti on. For d = 1, both the macroscopic homogeneity condition and the diffusion coefficient can be read from an explicit expression for the Green's functi on. Except for this case, the two available formulas for the effective diff usion matrix kappa do not explicit show how macroscopic homogenization take s place. Using an electrostatic analogy due to Anshelevich, Khanin and Sina i [AKS], we discuss upper and lower bounds on the diffusion coefficient kap pa for d > 1.