  Variable-range hopping transport along short one-dimensional wires and across
the shortest dimension of thin three-dimensional films and narrow
two-dimensional ribbons is studied theoretically. Geometric and transport
characteristics of the hopping resistor network are shown to depend on
temperature T and the dimensionality of the system. In two and three dimensions
the usual Mott law applies at high T where the correlation length of the
network is smaller than the sample thickness. As T decreases, the network
breaks into sparse filamentary paths, while the Mott law changes to a different
T-dependence, which is derived using the percolation theory methods. In one
dimension, deviations from the Mott law are known to exist at all temperatures
because of rare fluctuations. The evolution of such fluctuations from
highly-resistive "breaks" at high T to highly-conducting "shorts" at low T is
elucidated.
