  It is shown that for the hopping regime, the thermopowers in both finite
two-terminal and three-terminal systems are governed by the edges of the
samples. This is due to the fact that the energy transfer between a transport
electron and a conducting terminal is determined by the site most strongly
coupled to that terminal. One-dimensional systems with both nearest-neighbor
and variable-range transport as well as certain types of two-dimensional
systems, are considered. For a given sample, the changes in the thermopowers
due to modifying the bulk are quite limited, compared with those of the
conductance. When the small thermopower changes exist, their average over a
large ensemble of mesoscopic samples will vanish. We also obtain the
distribution of the thermopower in such an ensemble and show that its width
approaches a finite limit with increasing sample length. This contrasts with
the distribution of conductances in such systems, whose width vanishes in the
long sample limit. Finally, we find that the thermal conductances in the
three-terminal case have a boundary-dominated contribution, due to
non-percolating conduction paths. This contribution can become dominant when
the usual conductance is small enough. All our theoretical statements are
backed by numerical computations.
