Comparative study of spanning cluster distributions in different dimensions

The probability distributions of the masses of the clusters spanning from top to bottom of a percolating lattice at the percolation threshold are obtained in all dimensions from two to five. The first two cumulants and the exponents for the universal scaling functions are shown to have simple power law variations with the dimensionality. The cases where multiple spanning clusters occur are discussed separately and compared.Comment: 8 pages, latex, 4 eps figures included, to appear in Int. Journal of Modern Physics

Probability of Incipient Spanning Clusters in Critical Square Bond Percolation

The probability of simultaneous occurence of at least k spanning clusters has been studied by Monte Carlo simulations on the 2D square lattice at the bond percolation threshold $p_c=1/2$. It is found that the probability of k and more Incipient Spanning Clusters (ISC) has the values $P(k>1) \approx 0.00658(3)$ and $P(k>2) \approx 0.00000148(21)$ provided that the limit of these probabilities for infinite lattices exists. The probability $P(k>3)$ of more than three ISC could be estimated to be of the order of 10^{-11} and is beyond the possibility to compute a such value by nowdays computers. So, it is impossible to check in simulations the Aizenman law for the probabilities when $k>>1$. We have detected a single sample with 4 ISC in a total number of about 10^{10} samples investigated. The probability of single event is 1/10 for that number of samples.Comment: 7 pages, 1 table, 5 figures (1PS+4*Latex),uses epsf.sty Int.J.Mod.Phys. C (submitted to

Effects of boundary conditions on the critical spanning probability

The fractions of samples spanning a lattice at its percolation threshold are found by computer simulation of random site-percolation in two- and three-dimensional hypercubic lattices using different boundary conditions. As a byproduct we find $p_c = 0.311605(5)$ in the cubic lattice.Comment: 8 pages Latex, To appear in Int. J. Mod. Phys.

Fractal geometry of critical Potts clusters

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. However, the approach to the asymptotic region is slow, and the present range of the data does not allow a unique identification of the exact correction exponentsComment: 7 pages, 8 figures, Late

Are the Tails of Percolation Thresholds Gaussians ?

The probability distribution of percolation thresholds in finite lattices were first believed to follow a normal Gaussian behaviour. With increasing computer power and more efficient simulational techniques, this belief turned to a stretched exponential behaviour, instead. Here, based on a further improvement of Monte Carlo data, we show evidences that this question is not yet answered at all.Comment: 7 pages including 3 figure

Exact results at the 2-D percolation point

We derive exact expressions for the excess number of clusters b and the excess cumulants b_n of a related quantity at the 2-D percolation point. High-accuracy computer simulations are in accord with our predictions. b is a finite-size correction to the Temperley-Lieb or Baxter-Temperley-Ashley formula for the number of clusters per site n_c in the infinite system limit; the bn correct bulk cumulants. b and b_n are universal, and thus depend only on the system's shape. Higher-order corrections show no apparent dependence on fractional powers of the system size.Comment: 12 pages, 2 figures, LaTeX, submitted to Physical Review Letter

Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at occupation probability 0.59274621(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.Comment: 8 pages, including 3 postscript figures, minor corrections in this version, plus updated figures for the position of the percolation transitio

Transport on percolation clusters with power-law distributed bond strengths: when do blobs matter?

The simplest transport problem, namely maxflow, is investigated on critical percolation clusters in two and three dimensions, using a combination of extremal statistics arguments and exact numerical computations, for power-law distributed bond strengths of the type $P(\sigma) \sim \sigma^{-\alpha}$. Assuming that only cutting bonds determine the flow, the maxflow critical exponent \ve is found to be \ve(\alpha)=(d-1) \nu + 1/(1-\alpha). This prediction is confirmed with excellent accuracy using large-scale numerical simulation in two and three dimensions. However, in the region of anomalous bond capacity distributions ($0\leq \alpha \leq 1$) we demonstrate that, due to cluster-structure fluctuations, it is not the cutting bonds but the blobs that set the transport properties of the backbone. This ``blob-dominance'' avoids a cross-over to a regime where structural details, the distribution of the number of red or cutting bonds, would set the scaling. The restored scaling exponents however still follow the simplistic red bond estimate. This is argued to be due to the existence of a hierarchy of so-called minimum cut-configurations, for which cutting bonds form the lowest level, and whose transport properties scale all in the same way. We point out the relevance of our findings to other scalar transport problems (i.e. conductivity).Comment: 9 pages + Postscript figures. Revtex4+psfig. Submitted to PR

Statistical properties of the low-temperature conductance peak-heights for Corbino discs in the quantum Hall regime

A recent theory has provided a possible explanation for the ``non-universal scaling'' of the low-temperature conductance (and conductivity) peak-heights of two-dimensional electron systems in the integer and fractional quantum Hall regimes. This explanation is based on the hypothesis that samples which show this behavior contain density inhomogeneities. Theory then relates the non-universal conductance peak-heights to the ``number of alternating percolation clusters'' of a continuum percolation model defined on the spatially-varying local carrier density. We discuss the statistical properties of the number of alternating percolation clusters for Corbino disc samples characterized by random density fluctuations which have a correlation length small compared to the sample size. This allows a determination of the statistical properties of the low-temperature conductance peak-heights of such samples. We focus on a range of filling fraction at the center of the plateau transition for which the percolation model may be considered to be critical. We appeal to conformal invariance of critical percolation and argue that the properties of interest are directly related to the corresponding quantities calculated numerically for bond-percolation on a cylinder. Our results allow a lower bound to be placed on the non-universal conductance peak-heights, and we compare these results with recent experimental measurements.Comment: 7 pages, 4 postscript figures included. Revtex with epsf.tex and multicol.sty. The revised version contains some additional discussion of the theory and slightly improved numerical result

Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold

The incipient infinite cluster appearing at the bond percolation threshold can be decomposed into singly-connected ``links'' and multiply-connected ``blobs.'' Here we decompose blobs into objects known in graph theory as 3-blocks. A 3-block is a graph that cannot be separated into disconnected subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and 3-blocks are special cases of $k$-blocks with $k=1$, 2, and 3, respectively. We study bond percolation clusters at the percolation threshold on 2-dimensional square lattices and 3-dimensional cubic lattices and, using Monte-Carlo simulations, determine the distribution of the sizes of the 3-blocks into which the blobs are decomposed. We find that the 3-blocks have fractal dimension $d_3=1.2\pm 0.1$ in 2D and $1.15\pm 0.1$ in 3D. These fractal dimensions are significantly smaller than the fractal dimensions of the blobs, making possible more efficient calculation of percolation properties. Additionally, the closeness of the estimated values for $d_3$ in 2D and 3D is consistent with the possibility that $d_3$ is dimension independent. Generalizing the concept of the backbone, we introduce the concept of a ``$k$-bone'', which is the set of all points in a percolation system connected to $k$ disjoint terminal points (or sets of disjoint terminal points) by $k$ disjoint paths. We argue that the fractal dimension of a $k$-bone is equal to the fractal dimension of $k$-blocks, allowing us to discuss the relation between the fractal dimension of $k$-blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when printe