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 $d\ge 2$, 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 random resistor network for v.r.h. of \cite[Section~IV]{AHL} 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.