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

TL;DR

This paper proves that the infinite volume conductivity of the Miller-Abrahams resistor network converges to an effective homogenized matrix, linking it to Mott random walk diffusion and extending bounds consistent with Mott's law.

Contribution

It introduces a 2-scale homogenization approach to characterize the conductivity of the Miller-Abrahams network and relates it to the Mott random walk diffusion matrix.

Findings

The infinite volume conductivity is given by an effective homogenized matrix D.

Matrix D admits a variational characterization.

Results extend to models like the random conductance model without ellipticity assumptions.

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.