934 research outputs found

    Percolation on dual lattices with k-fold symmetry

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    Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p=1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particular, it enables us to answer a question of Grimmett concerning the anisotropic random cluster model on the triangular lattice.Comment: 11 pages, 1 figure. Revised with applications added; to appear in Random Structures and Algorithm

    Rigorous confidence intervals for critical probabilities

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    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

    Some Exact Results on Bond Percolation

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    We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice Λ\Lambda by ℓ\ell bonds connecting the same adjacent vertices, thereby yielding the lattice Λℓ\Lambda_\ell. This relation is used to calculate the bond percolation threshold on Λℓ\Lambda_\ell. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d≥2d \ge 2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N→∞N \to \infty limits of several families of NN-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N→∞N \to \infty.Comment: 33 pages latex 3 figure

    Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions

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    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
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