717 research outputs found

    Rigorous confidence intervals for critical probabilities

    Full text link
    We use the method of Balister, Bollobas and Walters to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases

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

    Full text link
    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

    Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions

    Full text link
    We extend the method of Balister, Bollob\'as and Walters for determining rigorous confidence intervals for the critical threshold of two dimensional lattices to three (and higher) dimensional lattices. We describe a method for determining a full confidence interval and apply it to show that the critical threshold for bond percolation on the simple cubic lattice is between 0.2485 and 0.2490 with 99.9999% confidence, and the critical threshold for site percolation on the same lattice is between 0.3110 and 0.3118 with 99.9999% confidence

    On the Number of Incipient Spanning Clusters

    Full text link
    In critical percolation models, in a large cube there will typically be more than one cluster of comparable diameter. In 2D, the probability of k>>1k>>1 spanning clusters is of the order e−αk2e^{-\alpha k^{2}}. In dimensions d>6, when η=0\eta = 0 the spanning clusters proliferate: for L→∞L\to \infty the spanning probability tends to one, and there typically are ≈Ld−6 \approx L^{d-6} spanning clusters of size comparable to |\C_{max}| \approx L^4. The rigorous results confirm a generally accepted picture for d>6, but also correct some misconceptions concerning the uniqueness of the dominant cluster. We distinguish between two related concepts: the Incipient Infinite Cluster, which is unique partly due to its construction, and the Incipient Spanning Clusters, which are not. The scaling limits of the ISC show interesting differences between low (d=2) and high dimensions. In the latter case (d>6 ?) we find indication that the double limit: infinite volume and zero lattice spacing, when properly defined would exhibit both percolation at the critical state and infinitely many infinite clusters.Comment: Latex(2e), 42 p, 5 figures; to appear in Nucl. Phys. B [FS

    Expansion in high dimension for the growth constants of lattice trees and lattice animals

    Full text link
    We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.Comment: 38 pages, 8 figures. Added section 6 to obtain the first term in the expansion, making the present paper more self-contained with very little change to the structure of the original paper. Accepted for publication in Combinatorics Probability and Computin

    Directed percolation near a wall

    Full text link
    Series expansion methods are used to study directed bond percolation clusters on the square lattice whose lateral growth is restricted by a wall parallel to the growth direction. The percolation threshold pcp_c is found to be the same as that for the bulk. However the values of the critical exponents for the percolation probability and mean cluster size are quite different from those for the bulk and are estimated by β1=0.7338±0.0001\beta_1 = 0.7338 \pm 0.0001 and γ1=1.8207±0.0004\gamma_1 = 1.8207 \pm 0.0004 respectively. On the other hand the exponent Δ1=β1+γ1\Delta_1=\beta_1 +\gamma_1 characterising the scale of the cluster size distribution is found to be unchanged by the presence of the wall. The parallel connectedness length, which is the scale for the cluster length distribution, has an exponent which we estimate to be ν1∥=1.7337±0.0004\nu_{1\parallel} = 1.7337 \pm 0.0004 and is also unchanged. The exponent τ1\tau_1 of the mean cluster length is related to β1\beta_1 and ν1∥\nu_{1\parallel} by the scaling relation ν1∥=β1+τ1\nu_{1\parallel} = \beta_1 + \tau_1 and using the above estimates yields τ1=1\tau_1 = 1 to within the accuracy of our results. We conjecture that this value of τ1\tau_1 is exact and further support for the conjecture is provided by the direct series expansion estimate τ1=1.0002±0.0003\tau_1= 1.0002 \pm 0.0003.Comment: 12pages LaTeX, ioplppt.sty, to appear in J. Phys.

    Percolation of even sites for random sequential adsorption

    Get PDF
    Consider random sequential adsorption on a red/blue chequerboard lattice with arrivals at rate 11 on the red squares and rate λ\lambda on the blue squares. We prove that the critical value of λ\lambda, above which we get an infinite blue component, is finite and strictly greater than 11.Comment: 35 pages. 13 figure
    • …
    corecore