Super-Instantons, Perfect Actions, Finite Size Scaling and the Continuum Limit

We discuss some aspects of the continuum limit of some lattice models, in particular the 2D O(N) models. The continuum limit is taken either in an infinite volume or in a box whose size is a fixed fraction of the infinite volume correlation length. We point out that in this limit the fluctuations of the lattice variables must be O(1) and thus restore the symmetry which may have been broken by the boundary conditions (b.c.). This is true in particular for the so-called super-instanton b.c. introduced earlier by us. This observation leads to a criterion to assess how close a certain lattice simulation is to the continuum limit and can be applied to uncover the true lattice artefacts, present even in the socalled ’perfect actions’. It also shows that David’s recent claim that super-instanton b.c. require a different renormalization must either be incorrect or an artefact of perturbation theory.