The author investigates the use of the phenomenological renormalisation group transformation in the nearest-neighbour square Ising model by means of correlation lengths defined by the moments of the pair correlation function.
Real space renormalisation group techniques and the replica method of Edwards and Anderson (1975), combined with successive variational approximations going beyond mean field, are used to calculate phase diagrams of various Ising systems. For a bond dilute 2D Ising model with nearest-neighbour interactions, real-space renormalisation group techniques provide very precise results which are used to test the reliability of the replica method. Combined with variational approximations going beyond mean field, this last method is applied to 3D Ising models with competing nearest-neighbour and next-nearest-neighbour interactions.
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.