  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 in dimension $d\geq 2$,
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
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 on a heuristic level.
Our proof uses also our previous results of stochastic homogenization and
percolation for the above models. The percolation results, and therefore also
our derivation of Mott's law, are exhaustive when the energy marks have a fixed
sign. For energy marks with different signs we have derived the low temperature
behavior of the critical conductance, while the proof of Mott's law still works
if the expected percolation property of left-right crossings in the
supercritical phase is assumed as an Ansatz.
