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Journal of Physics A Mathematical and General

General position-space renormalisation group for correlated percolation

G. F. Tuthill, W. Klein

Published Jul 1, 1982• 18 citations

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.

DOI: 10.1088/0305-4470/15/7/010

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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.

The author has investigated percolation properties of unfrustrated plaquettes in the two-dimensional Ising model with nearest-neighbour bonds +or-J distributed at random. The effects of plaquette correlations are considered using a two-parameter position-space renormalisation group (PSRG) approach, and it is found (within the framework of this approximate PSRG) that this correlated site percolation problem is in the same universality class as random site percolation. The percolation threshold is then found from large-cell PSRG methods to be indistinguishable from that for randomly placed sites.

Treats a percolation model with anisotropic bond occupation probabilities in order to study the crossover behaviour of the effective spatial dimension of the system. Previous position-space renormalisation group (PSRG) studies of this problem on the square lattice show a crossover from two-dimensional to pseudo-one-dimensional critical behaviour. To investigate the possible reasons for this surprising result, which is the opposite of the behaviour observed in thermal critical phenomena, the authors first develop alternative PSRG schemes. They study the predictions of these groups for asymptotically large rescaling parameters where the approximations in the PSRG are thought to become negligible. From a consideration of the approximations involved, they are led to a decimation transformation that uses an anisotropic cluster. The results of this method indicate that anisotropic percolation is in the same universality class as isotropic percolation, in complete analogy with thermal critical phenomena.

A position space renormalisation group technique (decimation) is used to treat site- and bond-diluted Ising ferromagnets on square and triangular lattices. In every case the renormalisation equations lead to two fixed points, one corresponding to the pure Ising system and one to percolation. Linearising around the fixed points, eigenvalues and hence the critical exponents nu T , nu P (correlation length), alpha (specific heat) and phi (crossover) are obtained. Numerical iteration of the recursion relations provides the critical curves. Values of critical temperatures, critical concentrations, exponents, and of limiting slopes of critical curves are in very satisfactory agreement with all known exact or series results. Points on the critical curve flow into the Ising fixed point under the transformation. The sign of the pure Ising exponent alpha is consistent with this flow for the site-diluted cases but not for the bond-diluted cases.

A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.