Miller-Abrahams random resistor network, Mott random walk and 2-scale homogenization

Abstract

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.