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Journal de Physique Lettres

Transfer-matrix approach to percolation and phenomenological renormalization

Bernard Derrida, J. Vannimenus

Published Jan 1, 1980• 68 citations
Publisher

Abstract

A transfer-matrix method is used to calculate the correlation length for strips of finite width in the bond and site percolation problem. From the knowledge of these correlation lengths we compute the thresholds and the critical exponent ν by the phenomenological renormalization method.
DOI
10.1051/jphyslet:019800041020047300

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A transfer-matrix simulation scheme for the three-dimensional (d=3) bond percolation is presented. Our scheme is based on Novotny's transfer-matrix formalism, which enables us to consider arbitrary (integral) number of sites N constituting a unit of the transfer-matrix slice even for d=3. Such an arbitrariness allows us to perform systematic finite-size-scaling analysis of the criticality at the percolation threshold. Diagonalizing the transfer matrix for N=4, 5,..., we obtain an estimate for the correlation-length critical exponent v=0.81(5).
A transfer-matrix-scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalization procedures are then used to analyze aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration p close to pc{\mathit{p}}_{\mathit{c}}pc​, the extent of exponential behavior of the Tc{\mathit{T}}_{\mathit{c}}Tc​\ifmmode\times\else\texttimes\fi{}p curve is clearly seen for over two decades of variation of p-pc{\mathit{p}}_{\mathit{c}}pc​. Close to the pure-system limit, the exactly known reduced slope is reproduced to a very good approximation, though with nonmonotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.
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.
The two-dimensional site percolation problem is studied by transfer-matrix methods on finite-width strips with free boundary conditions. The relationship between correlation length amplitudes and critical indices, predicted by conformal invariance, allows a very precise determination of the surface decay-of-correlations exponent, eta s =0.6664+or-0.0008, consistent with the analytical value eta s =2/3. It is found that a special transition does not occur in the case, corroborating earlier series results. At the ordinary transition, numerical estimates are consistent with the exact value y s =-1 for the irrelevant exponent.