© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical review. B, Condensed matter

Percolation with first- and second-neighbor bonds: Renormalization-group calculation of critical exponents

Rosane Riera, P.M.C. de Oliveira, Carlos Mauricio Chaves, S. L. A. de Queiroz

Published Oct 1, 1980• 13 citations

Abstract

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.

DOI: 10.1103/physrevb.22.3481

Related papers

Site Percolation in a Square Lattice with Second- and Dilute First-Neighbor Bonds

P.M.C. de Oliveira

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration $p$ of active first-neighbor bonds, and the concentration $q$ of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

Physical Review Letters•Nov 16, 1981•11 citations

A real-space renormalization group for site and bond percolation

PETER J. REYNOLDS, Eugene Stanley, W. Klein

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.

Journal of Physics C Solid State Physics•Apr 28, 1977•270 citations

Two- and three-dimensional site-bond-correlated percolation

N. S. Branco, Kleber D. Machado

We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.

Physical review. B, Condensed matter•Jan 1, 1993•10 citations

Renormalization Group Approach for Critical Percolation Behavior in a Two Dimensional Bond Percolating Square Lattice

Chiaki Murase

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 \(b=5/\sqrt{2}\) and \(6/\sqrt{2}\) 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.

Journal of the Physical Society of Japan•Apr 1, 1981•0 citations

Renormalization-group approach for critical percolation behavior in two dimensions

Y Yuge

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.

Physical review. B, Condensed matter•Aug 1, 1978•17 citations

First- and second-neighbor square-lattice Potts model: Renormalization-group treatment

P.M.C. de Oliveira, Constantino Tsallis, Georges Schwachheim

Within a real-space renormalization-group framework, we study the $q$-state Potts model in a square lattice, assuming a positive second-neighbor coupling constant and an arbitrary sign for the first-neighbor coupling constant. The $q$ evolution of the full phase diagram as well as the thermal and magnetic critical exponents are obtained. Whenever comparison with available results is possible, the agreement is either exact or very satisfactory.

Physical review. B, Condensed matter•Mar 1, 1984•15 citations

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.

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration

p

of active first-neighbor bonds, and the concentration

q

of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

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.

We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.

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/\sqrt{2}

and

6/\sqrt{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 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 renormalization-group framework, we study the

q

-state Potts model in a square lattice, assuming a positive second-neighbor coupling constant and an arbitrary sign for the first-neighbor coupling constant. The

b=5/\sqrt{2}

evolution of the full phase diagram as well as the thermal and magnetic critical exponents are obtained. Whenever comparison with available results is possible, the agreement is either exact or very satisfactory.