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Acta Physica Sinica

PERCOLATION WITH 3-BODY OR NEXT NEAREST NEIGHBOR CORRELATIONS

Tiecheng Li, 中国科学院物理研究所

Published Jan 1, 1982• 0 citations
Publisher

Abstract

The percolation problem for the 2-D triangular lattice of the quenched-correlated system is studied by using real-space renormalization group method. The many-body correlations for the occupied sites are considered by Kirkwood's superposition approximation approach and those for occupied and unoccupied sites by a procedure which holds the conservation of probability. The scheme is applied to the case including 3-body correlation and to that with next nearest neighboring correlations. Our conclusion is that, in either case the universality in critical phenomena of percolation ia unchanged, even some complicated correlations are involved.

DOI
10.7498/aps.31.1431

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The percolation problem for the 2-D triangular lattice of the quenched-correlated system is studied by using real-space renormalization group method. The many-body correlations for the occupied sites are considered by Kirkwood's superposition approximation approach and those for occupied and unoccupied sites by a procedure which holds the conservation of probability. The scheme is applied to the case including 3-body correlation and to that with next nearest neighboring correlations. Our conclusion is that, in either case the universality in critical phenomena of percolation ia unchanged, even some complicated correlations are involved.
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.
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We present a renormalization-group approach for treating the percolation properties of the nearest-neighbor triangular Ising model. We obtain exponents for the line of percolation transitions ${T}_{c}<~T<~\ensuremath{\infty}$. In particular, we find a possibly exact result for the connectedness-length exponent ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}$ in the "pure" percolation ($T=\ensuremath{\infty}$) limit, which holds for ${T}_{c}<T<~\ensuremath{\infty}$. At T=TcT={T}_{c}T=Tc​ we find the connectedness-length exponents of the percolation problem to be identical to the correlation-length exponents for the thermal problem.
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.