General position-space renormalisation group for correlated percolation

Abstract

A position-space renormalisation group (PSRG) approach has been developed for general site percolation problems in which the site occupancies are correlated, rather than independent, random variables. PSRG parameters are systematically introduced to describe multiple-site correlations. At the two- and four-site levels of approximation in the square lattice the authors find only two physical fixed points. One characterises random, or pure, percolation and has a universality class which is shown to encompass a variety of locally correlated site problems, including unfrustrated plaquettes and four-coordinated sites in a random-bond model. The second describes 'Ising-correlated' percolation and at the two-site level yields an excellent estimate (to within 2%) for the nearest-neighbour spin correlation function of the Ising model.