  We study a classical spin model (more precisely a class of models) with O(N)
symmetry that can be viewed as a simplified $D$ dimensional lattice model. It
is equivalent to a non-translationinvariant one dimensional model and contains
the dimensionality $D$ as a parameter that need not be an integer. The critical
dimension turns out to be 2, just as in the usual translation invariant models.
We study the phase structure, critical phenomena and spontaneous symmetry
breaking. Furthermore we compute the perturbation expansion to low order with
various boundary conditions. In our simplified models a number of questions can
be answered that remain controversial in the translation invariant models, such
as the asymptoticity of the perturbation expansion and the role of
super-instantons. We find that perturbation theory produces the right
asymptotic expansion in dimension $D\leq 2$ only with special boundary
conditions. Finally the model allows a test of the percolation ideas of
Patrascioiu and Seiler.
