632 research outputs found

### Percolation in Networks with Voids and Bottlenecks

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 $(3,12^2)$ 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 $p_c$ 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

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
$[0,1]$ 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:
$p_c=0.524430...$, $(3,12^2): p_c=0.740423...$, $(3^3,4^2): p_c=0.419615...$,
$(3,4,6,4):p_c=0.524821...$, $(4,8^2):p_c=0.676835...$, $(3^2,4,3,4)$:
$p_c=0.414120...$ . The results are generally within $10^{-5}$ of numerical
estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in
the formulas (if they are not exact) are less than $10^{-6}$.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

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
$tau$ (2.189(2)), the finite-size correction exponent $omega$ (0.64(2)), and
the scaling function exponent $sigma$ (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

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

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

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

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

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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