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Physical review. B, Condensed matter

Anisotropic bond percolation by position-space renormalization group

P.M.C. de Oliveira

Published Feb 1, 1982• 24 citations
Publisher

Abstract

We present a position-space renormalization-group procedure for the anisotropic bond-percolation problem in a square lattice. We use a kind of cell which preserves the geometrical features of the whole lattice, including duality. In this manner, the whole phase diagram and the dimensionality crossover exponent (both are exactly known) are reproduced for any scaling factor.

DOI
10.1103/physrevb.25.2034

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Physical review. B, Condensed matter•Jan 1, 1982•13 citations
We present a position-space renormalization-group procedure for the anisotropic bond-percolation problem in a square lattice. We use a kind of cell which preserves the geometrical features of the whole lattice, including duality. In this manner, the whole phase diagram and the dimensionality crossover exponent (both are exactly known) are reproduced for any scaling factor.
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.
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 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.
We propose a real-space renormalization-group approach for the bond-percolation problem in a square lattice with first- and second-neighbor bonds. We treat the respective probabilities as independent variables. Two types of cells are constructed. In one of them we consider the lattice as two interpenetrating sublattices, first-neighbor bonds playing the role of intersublattice links. This allows the calculation of both critical exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$, without resorting to any external field. Values found for the critical indices are in good agreement with data available in the literature. The phase diagram in parameter space is also obtained in each case.
We have determined the critical parameters for bond percolation on the square lattice using the recently developed large-cell renormalization-group method of Reynolds, Stanley, and Klein. The self-dual symmetry of our cells, which they share with the lattice itself, allows considerable simplification of the method and leads to rapid convergence toward the infinite-cell limit. We find that the scaling powers are given by ${y}_{p}=0.750\ifmmode\pm\else\textpm\fi{}0.005$ and ${y}_{h}=1.90\ifmmode\pm\else\textpm\fi{}0.017$. This leads, via scaling, to critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.667\ifmmode\pm\else\textpm\fi{}0.018$, $\ensuremath{\beta}=0.133\ifmmode\pm\else\textpm\fi{}0.023$, $\ensuremath{\gamma}=2.400\ifmmode\pm\else\textpm\fi{}0.048$, $\ensuremath{\delta}=19.0\ifmmode\pm\else\textpm\fi{}3.4$, $\ensuremath{\nu}=1.333\ifmmode\pm\else\textpm\fi{}0.009$, and $\ensuremath{\eta}=0.200\ifmmode\pm\else\textpm\fi{}0.034$.