934 research outputs found
Percolation on dual lattices with k-fold symmetry
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
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
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 by bonds connecting the same adjacent
vertices, thereby yielding the lattice . This relation is used to
calculate the bond percolation threshold on . We show that this
bond inflation leaves the universality class of the percolation transition
invariant on a lattice of dimensionality 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
limits of several families of -vertex graphs. Finally, we explore the effect
of bond vacancies on families of graphs with the property of bounded diameter
as .Comment: 33 pages latex 3 figure
Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions
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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