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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Multiscaling behavior in the crossover between surface and bulk critical exponents for percolation in two dimensions

Roberto Monetti, Ezequiel Vicente Albano

Published Jan 1, 1994• 3 citations
Publisher

Abstract

The crossover between the surface and bulk values of the order-parameter critical exponent (\ensuremath{\beta}) is investigated for the site-percolation model in a rectangular geometry (L\ifmmode\times\else\texttimes\fi{}M, L\ensuremath{\ll}M) with free boundary conditions. Due to the reduction in the connectivity, when the space is cut by the surfaces of the sample which cannot be penetrated by the percolating clusters, \ensuremath{\beta} decays exponentially from the surface value (${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$\ensuremath{\simeq}0.40) to the bulk value (${\mathrm{\ensuremath{\beta}}}_{\mathit{b}}$=5/36). This decay is interpreted in terms of multiscaling which implies that each row of the percolating cluster (taken in the M direction) has its own scaling and consequently a different fractal dimension.

DOI
10.1103/physreve.49.199

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The crossover between the surface and bulk values of the order-parameter critical exponent (\ensuremath{\beta}) is investigated for the site-percolation model in a rectangular geometry (L\ifmmode\times\else\texttimes\fi{}M, L\ensuremath{\ll}M) with free boundary conditions. Due to the reduction in the connectivity, when the space is cut by the surfaces of the sample which cannot be penetrated by the percolating clusters, \ensuremath{\beta} decays exponentially from the surface value (\ensuremathβs{\mathrm{\ensuremath{\beta}}}_{\mathit{s}}\ensuremathβs​\ensuremath{\simeq}0.40) to the bulk value (\ensuremathβb{\mathrm{\ensuremath{\beta}}}_{\mathit{b}}\ensuremathβb​=5/36). This decay is interpreted in terms of multiscaling which implies that each row of the percolating cluster (taken in the M direction) has its own scaling and consequently a different fractal dimension.
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.
A site-percolation model with two different sizes of particles is newly introduced on a square lattice. To estimate the fractal dimension and critical exponents with high accuracy, a finite-size scaling analysis is performed with a Monte Carlo simulation. The obtained fractal dimension coincides with that of the ordinary model, while the estimated exponents β, γ, and ν are slightly different from those of the ordinary model. However, the normalized exponents β^=β/ν\hat{\beta} = \beta / \nuβ^​=β/ν and γ^=γ/ν\hat{\gamma} = \gamma / \nuγ^​=γ/ν remain the same as in the ordinary model, which is a manifestation of weak universality.
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.
The critical exponents which describe quantities which are measured per unit 'mass' of the infinite cluster near percolation are shown to be shifted by beta p (the exponent describing the probability of belonging to this cluster). The fractal dimensionality of the infinite cluster then replaces the Euclidean one in hyperscaling relations. The crossover exponent for the effects of random fields on dilute Ising models are zero temperature is then shown to be phi h= gamma p+ beta p. Similarly, that for random local concentrations is phi p= alpha p+ beta p.
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.
Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager's critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}=(187/96 and a percolative susceptibility exponent \ensuremath{\gamma}=(91/48, in good agreement with existing numerical estimates. This d\ifmmode\bar\else\textasciimacron\fi{} is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent yJ{\mathrm{y}}_{\mathrm{J}}yJ​=(13/24 controlling the crossover between clusters and droplets.