Hopping conductivity in granular disordered systems

We study the temperature dependence of conductivity for high-resistivity granular disordered systems in terms of the critical-path method. It is shown that the low-field conductivity exp(-ATα) with α=12 is obeyed over large temperature ranges with possible crossovers to α=14 at low temperatures and to α>12 at high temperatures. The temperatures at which the crossovers occur depend on the distribution of grain sizes. Comparisons are made with experiments and with other theoretical approaches.