  We study a model of quasiparticles on a two-dimensional square lattice
coupled to Gaussian distributed dynamical fields. The model describes
quasiparticles coupled to spin or charge fluctuations and is solved by a Monte
Carlo sampling of the molecular field distributions. The non-perturbative
solution is compared to various approximations based on diagrammatic
perturbation theory. When the molecular field correlations are sufficiently
weak, the diagrammatic calculations capture the qualitative aspects of the
quasiparticle spectrum. For a range of model parameters near the magnetic
boundary, we find that the quasiparticle spectrum is qualitatively different
from that of a Fermi liquid in that it shows a double peak structure, and that
the diagrammatic approximations we consider fail to reproduce, even
qualitatively, the results of the Monte Carlo calculations. This suggests that
the pseudogap induced by a coupling to antiferromagnetic fluctuations and the
spin-splitting of the quasiparticle peak induced by a coupling to ferromagnetic
spin-fluctuations lie beyond diagrammatic perturbation theory.
