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

Surface shape and local critical behavior: The percolation problem in two dimensions

Bertrand Berche, Jean‐Marc Debierre, Hans-Peter Eckle

Published Dec 1, 1994• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.50.4542

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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.
We use a histogram Monte Carlo simulation method to calculate the scaling functions of the existence probability E p and the percolation probability P of the site percolation model on square lattices with free and periodic boundary conditions. We find that different boundary conditions give quite different scaling functions near the critical region. However, they give the consistent critical point, critical exponents, and the thermodynamic order parameter from renormalization-group calculations. Similar results are found for other percolation models. The implications of our calculated results for some theoretical problems of current interest are discussed.
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 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.
We present a numerical study of two-dimensional random-bond Potts ferromagnets. The model is studied both below and above the critical value Qc=4, which discriminates between second- and first-order transitions in the pure system. Two geometries are considered, namely cylinders and square-shaped systems, and the critical behavior is investigated through conformal invariance techniques that were recently shown to be valid, even in the randomness-induced second-order phase transition regime Q>4. In the cylinder geometry, connectivity transfer matrix calculations provide a simple test to find the range of disorder amplitudes that is characteristic of the disordered fixed point. The scaling dimensions then follow from the exponential decay of correlations along the strip. Monte Carlo simulations of spin systems on the other hand are generally performed on systems of rectangular shape on the square lattice, but the data are then perturbed by strong surface effects. The conformal mapping of a semi-infinite system inside a square enables us to take into account boundary effects explicitly and leads to an accurate determination of the scaling dimensions. The techniques are applied to different values of Q in the range 3-64.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.
Conformal transformations can be used to obtain the order parameter for two-dimensional systems at criticality in finite geometries with fixed boundary conditions on a connected boundary. To the known examples of this class (such as the disk and the infinite strip) we contribute the case of a rectangle. We show that the order parameter profile for simply connected boundaries can be represented as a universal function (independent of the criticality model) raised to the power $\frac{1}{2}\ensuremath{\eta}.$ The universal function can be determined from the Gaussian model or equivalently a problem in two-dimensional electrostatics. We show that fitting the order parameter profile to the theoretical form gives an accurate route to the determination of $\ensuremath{\eta}.$ We perform numerical simulations for the Ising model and percolation for comparison with these analytic predictions, and apply this approach to the study of the planar rotor model.