Decimation in more than one dimension

We develop a formalism for performing real space renormalization group transformations of the “decimation type” using perturbation theory. The type of transformations beyond d = 1 is nontrivial even for free theories. We check the formalism on solvable case of O(N) symmetric Heisenberg chain. The transformation is particularly useful to study asymptotically free theories. Results for one class of such models, the d = 2, O(N) symmetric σ models (N ≥ 3) for decimation with scale factor η = 2 (when quarter of the points is left) are given as an example.