  The Miller-Abrahams (MA) random resistor network is given by a complete graph
on a marked simple point process with edge conductivities depending on the
marks and decaying exponentially in the edge length. As Mott random walk, it is
an effective model to study Mott variable range hopping in amorphous solids as
doped semiconductors. By using 2-scale homogenization we prove that a.s. the
infinite volume conductivity of the MA resistor network is given by an
effective homogenized matrix $D$. Moreover $D$ admits a variational
characterization and equals the limiting diffusion matrix of Mott random walk.
This result clarifies the relation between the two models and it also allows to
extend to the MA resistor network the existing bounds on $D$ in agreement with
the physical Mott law [12,14]. The latter concerns the low temperature
stretched exponential decay of conductivity in amorphous solids. The techniques
developed here can be applied to other models, as e.g. the random conductance
model [11], without ellipticity assumptions.
