  We study the 3D anisotropic Hubbard model on a cubic lattice with hopping
parameter $t$ in the $x$- and $y$-directions and a possibly different hopping
parameter $t_z$ in the $z$-direction; this model interpolates between the 2D
and 3D Hubbard models corresponding to the limiting cases $t_z=0$ and $t_z=t$,
respectively. We first derive all-order asymptotic expansions for the density
of states. Using these expansions and units such that $t=1$, we analyze how the
N\'eel temperature and the antiferromagnetic mean field depend on the coupling
parameter, $U$, and on the hopping parameter $t_z$. We derive asymptotic
formulas valid in the weak coupling regime, and we study in particular the
transition from the three-dimensional to the two-dimensional model as $t_z \to
0$. It is found that the asymptotic formulas are qualitatively different for
$t_z = 0$ (the two-dimensional case) and $t_z > 0$ (the case of nonzero hopping
in the $z$-direction). Our results show that certain universality features of
the three-dimensional Hubbard model are lost in the limit $t_z \to 0$ in which
the three-dimensional model reduces to the two-dimensional model.
