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Journal of Physics A Mathematical and General

Finite-size scaling for directed bond percolation with and without cycles on a triangular lattice

Anna Chame, S. L. A. de Queiroz, Raimundo R. dos Santos

Published Jun 21, 1986• 0 citations

Abstract

The authors apply finite-size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic, directed cyclic and undirected bond percolation on a triangular lattice. The results are in good agreement with known estimates, and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases, as well as the distinction between locally directed and fully isotropic problems. In addition, the authors draw tentative comments on the extension, to directed problems, of the relation A= pi eta (where A is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point), known to hold for various isotropic systems.

DOI: 10.1088/0305-4470/19/9/012

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The authors apply finite-size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic, directed cyclic and undirected bond percolation on a triangular lattice. The results are in good agreement with known estimates, and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases, as well as the distinction between locally directed and fully isotropic problems. In addition, the authors draw tentative comments on the extension, to directed problems, of the relation A= pi eta (where A is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point), known to hold for various isotropic systems.

The correlation length of boundary spins in the Ising model, defined on strips of triangular lattice with free boundary conditions, is determined with an efficient numerical procedure based on the star-triangle transformation. In the case of isotropic critical interactions, the extrapolated amplitude of the correlation length is in excellent agreement with the value 2/( pi eta /sub ///) predicted by conformal invariance. An analytical formula for the amplitude in strips with anisotropic interactions is proposed. Fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor 1 / 2 . The convergence of phenomenological renormalisation with free boundary conditions is studied.

Percolation with bonds oriented along an external field direction is considered. Assuming that the second-order type phase transition picture can be applied in the vicinity of the critical point the threshold probability and the correlation length exponent nu 0 is calculated on the square lattice by Monte Carlo techniques using finite-size scaling arguments. It is found that nu 0 =1.65+or-0.06, i.e. the orientated percolation belongs to a different university class from the unorientated one. The critical probabilities for several two-dimensional lattices are determined.

A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.

Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.

The author proposes a new type of cluster in addition to those of Reynolds, Klein and Stanley for the square lattice and another in addition to those of Chao for the triangular lattice for a real-space renormalisation group approach for studying critical phenomena. Thus each lattice is associated with two types of clusters-the longitudinal and the diagonal. The results for the bond percolation and Ising fixed points obtained from the diagonal square and honeycomb clusters agree with the exact values to within 6-11% error; whereas those obtained from the longitudinal triangular and honeycomb clusters agree with the exact values to within 2-4.5% error. The critical exponents corresponding to the percolation and the Ising problems also show reasonably good results: in particular those calculated by using the longitudinal triangular cluster agree with the exact values to within 6.7% and 3.2% error respectively.