Renormalization of nonlocal degrees of freedom

The renormalization in real space of systems with nonlocal degrees of freedom is discussed and reviewed for the percolation model, for which the degrees of freedom, the clusters, are nonlocal and span all ranges in the system. Previous attempts, including recent ones, are critically examined and shown to lack a theoretical basis; in particular, they do not consider the partition function. It is demonstrated that, in contrast to the case of local degrees of freedom, all ranges of the degrees of freedom must be considered when eliminating nonlocal degrees of freedom at short distances. New couplings are naturally generated, and lead to a mapping between clusters in the original and renormalized systems, and thus to the renormalization-group equations. Remaining questions are discussed.