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

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