Corrections to scaling for percolative conduction: anomalous behavior at small L

Recently Grassberger has shown that the correction to scaling for the conductance of a bond percolation network on a square lattice is a nonmonotonic function of the linear lattice dimension with a minimum at $L = 10$, while this anomalous behavior is not present in the site percolation networks. We perform a high precision numerical study of the bond percolation random resistor networks on the square, triangular and honeycomb lattices to further examine this result. We use the arithmetic, geometric and harmonic means to obtain the conductance and find that the qualitative behavior does not change: it is not related to the shape of the conductance distribution for small system sizes. We show that the anomaly at small L is absent on the triangular and honeycomb networks. We suggest that the nonmonotonic behavior is an artifact of approximating the continuous system for which the theory is formulated by a discrete one which can be simulated on a computer. We show that by slightly changing the definition of the linear lattice size we can eliminate the minimum at small L without significantly affecting the large L limit.Comment: 3 pages, 4 figures;slightly expanded, 2 figures added. Accepted for publishing in Phys. Rev.

Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model

We study the dynamic behavior of the Ziff-Gulari-Barshad (ZGB) irreversible surface-reaction model around its kinetic second-order phase transition, using both epidemic and poisoning-time analyses. We find that the critical point is given by p_1 = 0.3873682 \pm 0.0000015, which is lower than the previous value. We also obtain precise values of the dynamical critical exponents z, \delta, and \eta which provide further numerical evidence that this transition is in the same universality class as directed percolation.Comment: REVTEX, 4 pages, 5 figures, Submitted to Physical Review

Precise determination of the bond percolation thresholds and finite-size scaling corrections for the s.c., f.c.c., and b.c.c. lattices

Extensive Monte-Carlo simulations were performed to study bond percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and body-centered cubic (b.c.c.) lattices, using an epidemic kind of approach. These simulations provide very precise values of the critical thresholds for each of the lattices: pc(s.c.) = 0.248 812 6(5), pc(f.c.c.) = 0.120 163 5(10), and pc(b.c.c.) = 0.180 287 5(10). For p close to pc, the results follow the expected finite-size and scaling behavior, with values for the Fisher exponent $tau$ (2.189(2)), the finite-size correction exponent $omega$ (0.64(2)), and the scaling function exponent $sigma$ (0.445(1)) confirmed to be universal.Comment: 16 pgs, 7 figures, LaTeX, to be published in Phys. Rev.

Scaling behavior of explosive percolation on the square lattice

Clusters generated by the product-rule growth model of Achlioptas, D'Souza, and Spencer on a two-dimensional square lattice are shown to obey qualitatively different scaling behavior than standard (random growth) percolation. The threshold with unrestricted bond placement (allowing loops) is found precisely using several different criteria based upon both moments and wrapping probabilities, yielding p_c = 0.526565 +/- 0.000005, consistent with the recent result of Radicchi and Fortunato. The correlation-length exponent nu is found to be close to 1. The qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability P_(infinity) (size of largest cluster), the susceptibility, and of the second moment of finite clusters, where discontinuities appears at the threshold. The critical cluster-size distribution does not follow a consistent power-law for the range of system sizes we study L 2 for larger L.Comment: v2: Updated results in original version with new data; expanded discussion. v3: Resubmitted version. New figures, reference

On the universality of distribution of ranked cluster masses at critical percolation

The distribution of masses of clusters smaller than the infinite cluster is evaluated at the percolation threshold. The clusters are ranked according to their masses and the distribution $P(M/L^D,r)$ of the scaled masses M for any rank r shows a universal behaviour for different lattice sizes L (D is the fractal dimension). For different ranks however, there is a universal distribution function only in the large rank limit, i.e., $P(M/L^D,r)r^{-y\zeta } \sim g(Mr^y/L^D)$ (y and $\zeta$ are defined in the text), where the universal scaling function g is found to be Gaussian in nature.Comment: 4 pages, to appear in J. Phys.

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

Kinetics of catalysis with surface disorder

We study the effects of generalised surface disorder on the monomer-monomer model of heterogeneous catalysis, where disorder is implemented by allowing different adsorption rates for each lattice site. By mapping the system in the reaction-controlled limit onto a kinetic Ising model, we derive the rate equations for the one and two-spin correlation functions. There is good agreement between these equations and numerical simulations. We then study the inclusion of desorption of monomers from the substrate, first by both species and then by just one, and find exact time-dependent solutions for the one-spin correlation functions.Comment: LaTex, 19 pages, 1 figure included, requires epsf.st

Recent advances and open challenges in percolation

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap, and correlated percolation

The Largest Cluster in Subcritical Percolation

The statistical behavior of the size (or mass) of the largest cluster in subcritical percolation on a finite lattice of size $N$ is investigated (below the upper critical dimension, presumably $d_c=6$). It is argued that as $N \to \infty$ the cumulative distribution function converges to the Fisher-Tippett (or Gumbel) distribution $e^{-e^{-z}}$ in a certain weak sense (when suitably normalized). The mean grows like $s_\xi^* \log N$, where $s_\xi^*(p)$ is a ``crossover size''. The standard deviation is bounded near $s_\xi^* \pi/\sqrt{6}$ with persistent fluctuations due to discreteness. These predictions are verified by Monte Carlo simulations on $d=2$ square lattices of up to 30 million sites, which also reveal finite-size scaling. The results are explained in terms of a flow in the space of probability distributions as $N \to \infty$. The subcritical segment of the physical manifold ($0 < p < p_c$) approaches a line of limit cycles where the flow is approximately described by a ``renormalization group'' from the classical theory of extreme order statistics.Comment: 16 pages, 5 figs, expanded version to appear in Phys Rev