632 research outputs found

    Percolation in Networks with Voids and Bottlenecks

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    A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the sub-net mesh size goes to zero. The validity of this method is tested for bond percolation on filled checkerboard and "stack-of-triangle" lattices. Thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method, while for the triangular system they are found exactly using the triangle-triangle transformation. The values of the thresholds approach the asymptotic values of 0.64222 and 0.53993 respectively as the mesh is made finer, consistent with a direct determination based upon the predicted critical corner-connection probability.Comment: to appear, Physical Review E. Small changes from first versio

    Predictions of bond percolation thresholds for the kagom\'e and Archimedean (3,122)(3,12^2) lattices

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    Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagom\'e and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2) bond problems, and the other gives estimates of pcp_c for the homogeneous kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.Comment: 4 pages, 5 figure

    Critical surfaces for general inhomogeneous bond percolation problems

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    We present a method of general applicability for finding exact or accurate approximations to bond percolation thresholds for a wide class of lattices. To every lattice we sytematically associate a polynomial, the root of which in [0,1][0,1] is the conjectured critical point. The method makes the correct prediction for every exactly solved problem, and comparison with numerical results shows that it is very close, but not exact, for many others. We focus primarily on the Archimedean lattices, in which all vertices are equivalent, but this restriction is not crucial. Some results we find are kagome: pc=0.524430...p_c=0.524430..., (3,122):pc=0.740423...(3,12^2): p_c=0.740423..., (33,42):pc=0.419615...(3^3,4^2): p_c=0.419615..., (3,4,6,4):pc=0.524821...(3,4,6,4):p_c=0.524821..., (4,82):pc=0.676835...(4,8^2):p_c=0.676835..., (32,4,3,4)(3^2,4,3,4): pc=0.414120...p_c=0.414120... . The results are generally within 10−510^{-5} of numerical estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in the formulas (if they are not exact) are less than 10−610^{-6}.Comment: Submitted to J. Stat. Mec

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

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    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 tautau (2.189(2)), the finite-size correction exponent omegaomega (0.64(2)), and the scaling function exponent sigmasigma (0.445(1)) confirmed to be universal.Comment: 16 pgs, 7 figures, LaTeX, to be published in Phys. Rev.

    Efficient Monte Carlo algorithm and high-precision results for percolation

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    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

    Finite-size scaling of the stochastic susceptible-infected-recovered model

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    The critical behavior of the stochastic susceptible-infected-recovered model on a square lattice is obtained by numerical simulations and finite-size scaling. The order parameter as well as the distribution in the number of recovered individuals is determined as a function of the infection rate for several values of the system size. The analysis around criticality is obtained by exploring the close relationship between the present model and standard percolation theory. The quantity UP, equal to the ratio U between the second moment and the squared first moment of the size distribution multiplied by the order parameter P, is shown to have, for a square system, a universal value 1.0167(1) that is the same as for site and bond percolation, confirming further that the SIR model is also in the percolation class

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

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    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

    Determination of the bond percolation threshold for the Kagome lattice

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    The hull-gradient method is used to determine the critical threshold for bond percolation on the two-dimensional Kagome lattice (and its dual, the dice lattice). For this system, the hull walk is represented as a self-avoiding trail, or mirror-model trajectory, on the (3,4,6,4)-Archimedean tiling lattice. The result pc = 0.524 405 3(3) (one standard deviation of error) is not consistent with the previously conjectured values.Comment: 10 pages, TeX, Style file iopppt.tex, to be published in J. Phys. A. in August, 199

    Exact bond percolation thresholds in two dimensions

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    Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.Comment: Accepted for publication in J. Phys
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