  We calculate discrete beta functions corresponding to the two-lattice
matching for the 2D O(N) models and Dyson's hierarchical model. We describe and
explain finite-size effects such as the appearance of a nontrivial infrared
fixed point that goes to infinity at infinite volume or the merging of an
infrared and an ultraviolet fixed point. We present extensions of the RG flows
to the complex coupling plane. We discuss the possibility of constructing a
continuous beta function from the discrete one by using functional conjugation
methods. We briefly discuss the relevance of these findings for the search of
nontrivial fixed points in multiflavor lattice gauge theory models.
