Mott’s law for the v.r.h. random resistor network and for Mott’s random walk

Mott’s variable range hopping (v.r.h.) is the phonon-induced hopping of electrons in disordered solids (such as doped semiconductors) within the regime of strong Anderson localization. It was introduced by N. Mott to explain the anomalous low temperature conductivity decay in dimension d ≥ 2, corresponding now to the so called Mott’s law. We provide a rigorous derivation of this Physics law for two effec- tive models of Mott v.r.h.: the random resistor network for v.r.h. of [1, Section IV] 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 on a heuristic level.