Mott's law for the Miller-Abrahams random resistor network and for Mott's random walk

Abstract

Mott's variable range hopping (v.r.h.) is the phonon-induced hopping of electrons in disordered solids (as doped semiconductors) within the regime of strong Anderson localization. It was introduced by N. Mott to explain the anomalous low temperature conductivity decay corresponding now to the so called Mott's law. We provide a rigorous derivation of this Physics law for two effective models of Mott v.r.h.: the Miller-Abrahams random resistor network and Mott's random walk. We also determine the constant multiplying the power of the inverse temperature in the exponent in Mott's law, which was an open problem also at heuristic level. Our proof uses also our previous results of homogenization and percolation for the above models, which are exhaustive when the energy marks have a fixed sign. For energy marks with different signs the expected percolation property of left-right crossings in the supercritical phase (on which we are currently working) is assumed as an Ansatz.