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arXiv:cond-mat/9506005 · Journal of Physics A Mathematical and General

On the surface properties of two-dimensional percolation clusters

S L A de Queiroz

Published Jul 7, 1995• 4 citations
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Abstract

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.

DOI
10.1088/0305-4470/28/13/002

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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.
We investigate the site-percolation problem on the square and simple-cubic lattices by means of a Monte Carlo algorithm that in fact simulates systems with size ${L}^{d\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, where LLL specifies the linear system size. This algorithm can be regarded either as an extension of the Hoshen-Kopelman method or as a special case of the transfer-matrix Monte Carlo technique. Various quantities, such as the magnetic correlation function, are sampled in the finite directions of the above geometry. Simulations are arranged such that both bulk and surface quantities can be sampled. On the square lattice, we locate the percolation threshold at pc=0.5927465(4){p}_{c}=0.592\phantom{\rule{0.2em}{0ex}}746\phantom{\rule{0.2em}{0ex}}5(4)pc​=0.5927465(4), and determine two universal quantities as Qgbc=0.93034(1){Q}_{gbc}=0.930\phantom{\rule{0.2em}{0ex}}34(1)Qgbc​=0.93034(1) and Qgsc=0.79372(3){Q}_{gsc}=0.793\phantom{\rule{0.2em}{0ex}}72(3)Qgsc​=0.79372(3), which are associated with bulk and surface correlations, respectively. These values agree well with the exact values ${2}^{\ensuremath{-}5∕48}$ and ${2}^{\ensuremath{-}1∕3}$, respectively, which follow from conformal invariance. On the simple-cubic lattice, we locate the percolation threshold at pc=0.3116077(4){p}_{c}=0.311\phantom{\rule{0.2em}{0ex}}607\phantom{\rule{0.2em}{0ex}}7(4)pc​=0.3116077(4). We further determine the bulk thermal and magnetic exponents as yt=1.1437(6){y}_{t}=1.1437(6)yt​=1.1437(6) and yh=2.5219(2){y}_{h}=2.5219(2)yh​=2.5219(2), respectively, and the surface magnetic exponent at the ordinary phase transition as yhs(o)=1.0248(3){y}_{hs}^{(\mathrm{o})}=1.0248(3)yhs(o)​=1.0248(3).
The critical behavior of site percolation confined to the interior of a parabola is investigated. At the percolation threshold, conformal mapping is used to derive analytical expressions for the pair correlation function and the local order parameter in the case of free and fixed boundary conditions, respectively. Numerical results obtained by intensive Monte Carlo simulations are in excellent agreement with the analytical expressions. The numerical data for the off-critical local order parameter with free boundary conditions show that correction to scaling terms are important and the leading correction exponent is estimated to be \ensuremath{\epsilon}\ensuremath{\simeq}0.18.
The two-dimensional random-bond Ising model is numerically studied on long strips by transfer-matrix methods. It is shown that the rate of decay of correlations at criticality, as derived from averages of the two largest Lyapunov exponents plus conformal invariance arguments, differs from that obtained through direct evaluation of correlation functions. The latter is found to be, within error bars, the same as in pure systems. Our results confirm field-theoretical predictions. The conformal anomaly c is calculated from the leading finite-width correction to the averaged free energy on the strips. Estimates thus obtained are consistent with c=1/2, the same as for the pure Ising model.
Using Monte Carlo methods and finite-size scaling, we investigate surface critical phenomena in the bond-percolation model on the simple-cubic lattice with two open surfaces in one direction. We decompose the whole lattice into percolation clusters and sample the surface and bulk dimensionless ratios Q1 and Qb, defined on the basis of the moments of the cluster-size distribution. These ratios are used to determine critical points. At the bulk percolation threshold pbc, we determine the surface bond-occupation probability at the special transition as p(s)1c = 0.418 17(2), and further obtain the corresponding surface thermal and magnetic exponents as y(s)t1 = 0.5387(2) and y(s)h1 = 1.8014(6), respectively. Next, from the pair correlation function on the surfaces, we determine y(o)h1 = 1.0246(4) and y(e)h1 = 1.25(6) for the ordinary and the extraordinary transition, respectively, of which the former is consistent with the existing result y(o)h1 = 1.024(4). We also numerically derive the line of surface phase transitions occurring at pb < pbc, and determine the pertinent asymptotic values of the universal ratios Q1 and Qb.
We consider the two-dimensional (2D) random Ising model on a diagonal strip of the square lattice, where the bonds take two values, J1>J2, with equal probability. Using an iterative method, based on a successive application of the star-triangle transformation, we have determined at the bulk critical temperature the correlation length along the strip xi(L) for different widths of the strip L</=21. The ratio of the two lengths xi(L)/L=A is found to approach the universal value A=2/pi for large L, independent of the dilution parameter J(1)/J(2). With our method we have demonstrated with high numerical precision, that the surface correlation function of the 2D dilute Ising model is self-averaging, in the critical point conformally covariant and the corresponding decay exponent is eta( ||)=1.