  In recent literature on trapped ultracold atomic gases, calculations for
2D-systems are often done within the Dynamical Mean Field Theory (DMFT)
approximation. In this paper, we compare DMFT to a fully two-dimensional,
self-consistent second order perturbation theory for weak interactions in a
repulsive Fermi-Hubbard model. We investigate the role of quantum and of
spatial fluctuations when the system is in the antiferromagnetic phase, and
find that, while quantum fluctuations decrease the order parameter and critical
temperatures drastically, spatial fluctuations only play a noticeable role when
the system undergoes a phase transition, or at phase boundaries in the trap. We
conclude from this that DMFT is a good approximation for the antiferromagnetic
Fermi-Hubbard model for experimentally relevant system sizes.
