  We present a comprehensive pedagogical discussion of a family of models
describing the propagation of a single particle in a multicomponent
non-Markovian Gaussian random field. We report some exact results for
single-particle Green's functions, self-energy, vertex part and T-matrix. These
results are based on a closed form solution of the Dyson equation combined with
the Ward identity. Analytical properties of the solution are discussed. Further
we describe the combinatorics of the Feynman diagrams for the Green's function
and the skeleton diagrams for the self-energy and vertex, using recurrence
relations between the Taylor expansion coefficients of the self-energy.
Asymptotically exact equations for the number of skeleton diagrams in the limit
of large N are derived. Finally, we consider possible realizations of a
multicomponent Gaussian random potential in quantum transport via complex
quantum dot experiments.
