Journal of Statistical Physics, Vol. 112, Nos. 3/4, August 2003. p. 457-495.
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-translationinvanant 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 \< 2 only with special boundary conditions. Finally the model allows a test of the percolation ideas of Patrascioiu and Seiler.