  In this paper, we solve two problems dealing with the homogenization of
random media. We show that a random quasiconformal mapping is close to an
affine mapping, while a circle packing of a random Delauney triangulation is
close to a conformal map, confirming a conjecture of Stephenson. We also show
that on a Riemann surface equipped with a conformal metric, a random Delauney
triangulation is close to being circle packed.
