717 research outputs found
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
Predictions of bond percolation thresholds for the kagom\'e and Archimedean lattices
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 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
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
In critical percolation models, in a large cube there will typically be more
than one cluster of comparable diameter. In 2D, the probability of
spanning clusters is of the order . In dimensions d>6, when
the spanning clusters proliferate: for the spanning
probability tends to one, and there typically are 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
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
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 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 and respectively. On the other hand the exponent
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 and is also unchanged. The exponent of the mean
cluster length is related to and by the scaling
relation and using the above estimates
yields to within the accuracy of our results. We conjecture that
this value of is exact and further support for the conjecture is
provided by the direct series expansion estimate .Comment: 12pages LaTeX, ioplppt.sty, to appear in J. Phys.
Percolation of even sites for random sequential adsorption
Consider random sequential adsorption on a red/blue chequerboard lattice with
arrivals at rate on the red squares and rate on the blue squares.
We prove that the critical value of , above which we get an infinite
blue component, is finite and strictly greater than .Comment: 35 pages. 13 figure
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