Relevance and limits of Mott's law in disordered insulators.

Twenty five years ago, Mott's law was established in order to describe electrical transport in disordered insulators at low temperature. In this review, we briefly summarize the different theoretical steps involved in the rigourous derivation of Mott's law. We stress upon the fact that Mott's law gives the mean conductance of an ensemble of macroscopic samples as long as electron-electron interactions remain negligible. We then study what happens when at least one of the key assumptions of Mott's law no longer holds. We first focus on systems whose size - at least in one dimension - is not macroscopic: the optimization involved in Mott's law is no longer relevant for the measured conductance. Eventually, we try to gather different works dealing with electron-electron interactions. It is now established that interactions generally produce a stronger divergence for the electrical resistance than the one predicted by Mott's law at the lowest temperatures. But the exact shape of this divergence, as well as its interpretation, remain debated. We try to make a link between Efros and Shklovskii's work and their famous ''Coulomb gap'' and a more recent work about granular media. In this latter work, the size of the grains is the key parameter for the shape of the divergence of the resistance at low temperature. We suggest this could indicate a way for a model accounting for the different shapes of divergence of the electrical resistance at the lowest temperatures, Furthermore this framework of granular media allows us to deal with non linear regime: we explain the main differences between the predictions of the hot electrons model and the ones recently derived for a d-dimensional network of grains.