  Using the tools of random matrix theory we develop a statistical analysis of
the transport properties of thermoelectric low-dimensional systems made of two
electron reservoirs set at different temperatures and chemical potentials, and
connected through a low-density-of-states two-level quantum dot that acts as a
conducting chaotic cavity. Our exact treatment of the chaotic behavior in such
devices lies on the scattering matrix formalism and yields analytical
expressions for the joint probability distribution functions of the Seebeck
coefficient and the transmission profile, as well as the marginal
distributions, at arbitrary Fermi energy. The scattering matrices belong to
circular ensembles which we sample to numerically compute the transmission
function, the Seebeck coefficient, and their relationship. The exact transport
coefficients probability distributions are found to be highly non-Gaussian for
small numbers of conduction modes, and the analytical and numerical results are
in excellent agreement. The system performance is also studied, and we find
that the optimum performance is obtained for half-transparent quantum dots;
further, this optimum may be enhanced for systems with few conduction modes.
