  We study a model of quasiparticles on a two-dimensional square lattice
coupled to Gaussian distributed dynamical molecular fields. The model describes
quasiparticles coupled to spin or charge fluctuations, and is solved by a Monte
Carlo sampling of the molecular field distributions. When the molecular field
correlations are sufficiently weak, the corrections to the self-consistent
Eliashberg theory do not bring about qualitative changes in the quasiparticle
spectrum. But for a range of model parameters near the magnetic boundary, we
find that Migdal's theorem does not apply and the quasiparticle spectrum is
qualitatively different from its mean-field approximation, in that a pseudogap
opens in the quasiparticle spectrum. An important feature of the magnetic
pseudogap found in the present calculations is that it is strongly anisotropic.
It vanishes anlong the diagonal of the Brillouin zone and is large near the
zone boundary. In the case of ferromagnetic fluctuations, we also find a range
of model parameters with qualitative changes in the quasiparticle spectral
function not captured by the one-loop approximation, in that the quasiparticle
peak splits into two. We provide intuitive arguments to explain the physical
origin of the breakdown of Midgal's theorem
