On some asymptotic formulas in the strong localization regime of the theory of disordered systems

We present certain results, both heuristic and rigorous, on the asymptotics study of the simplest interesting observables of the theory of disordered systems: the density of states (the DOS) and the conductivity. First we outline heuristic arguments dating back to I.Lifshitz and allowing us to write the leading term of the low-energy asymptotics of the logarithm of the DOS for the non-positive Poisson potential, less studied so far than, say, the alloy type potential and modeling chaotically distributed attractive impurities. Second we formulate recent rigorous results ([16]) that justify and develop this heuristics in the case of singular one-impurity potentials, the screened Coulomb in particular. At last we presents heuristic arguments ([9]) that, we believe, make more precise and detailed the arguments of I. Lifshitz and N. Mott and allowing us to obtain the known Mott formula for the low-frequency conductivity of disordered systems. The arguments are based on an Ansatz describing the form of relevant realizations of the random potential. We believe that the Ansatz is a natural consequence of the studies of the strong localization regime carried out so far and can be used to write a number of other observables in the strong localization regime.