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Physical review. B, Condensed matter

Anisotropic bond percolation in two dimensions

Daniel P. Arovas, R. N. Bhatt, Boris Shapiro

Published Aug 1, 1983• 6 citations
Publisher

Abstract

A new single-parameter renormalization-group equation is formulated for anisotropic bond percolation in two dimensions using a position-space renormalization approach. The new equation yields the exact critical line ${p}_{x}+{p}_{y}=1$ within both the Migdal-Kadanoff decimation and cell renormalization schemes. For large anisotropy, however, an additional critical line appears leading to a spurious divergence in the correlation-length critical exponent. An alternative scheme, which does not preserve the exact critical surface, but yields a correlation-length exponent relatively independent of anisotropy, is also examined.

DOI
10.1103/physrevb.28.1433

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A new single-parameter renormalization-group equation is formulated for anisotropic bond percolation in two dimensions using a position-space renormalization approach. The new equation yields the exact critical line px+py=1{p}_{x}+{p}_{y}=1px​+py​=1 within both the Migdal-Kadanoff decimation and cell renormalization schemes. For large anisotropy, however, an additional critical line appears leading to a spurious divergence in the correlation-length critical exponent. An alternative scheme, which does not preserve the exact critical surface, but yields a correlation-length exponent relatively independent of anisotropy, is also examined.
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 renormalization group approach in real space is applied to the bond percolating square lattice in two dimension using two scaling transformations proposed by Reynolds et al. (Reynolds's method) and by Yuge (Yuge's method). The renormalization transformation R b ( p ), the eigenvalue λ( b ) and the correlation length critical exponent v ( b ) are calculated exactly by changing the scale of length b =4 for Reynolds's method and b=5/2b=5/\sqrt{2}b=5/2​ and 6/26/\sqrt{2}6/2​ for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.
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.
In complete analogy with thermal critical phenomena, it is expected that anisotropic percolation is in the same universality class as the isotropic one. However the previous result, obtained from the closed form conventional cell real space renormalization group method on the square lattice, the isotropic fixed point is completely unstable is known. We examine this in the large cell limit by using a Monte Carlo renormalization group method and show that the scaling exponent associated with anisotropy is not relevant.
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.