Site percolation Square lattice 400 x 400 Three largest clusters are coloured green/blue/yellow
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Transcript of Site percolation Square lattice 400 x 400 Three largest clusters are coloured green/blue/yellow
Site percolation
Square lattice400 x 400
Three largest clusters are coloured green/blue/yellow
p = 0.50
p = 0.55
p = 0.58
p = 0.59
p = 0.60
p = 0.65
Fraction of sites on the largest cluster
S = mean size of finite clusters
Estimate in Bunde and Havlin: pc = 0.5927
Cluster generation via Leath method(epidemic spreading)
M ~ rdfr
space dimension d = 2
fractal dimension df
r
At pc the infinite cluster has a fractal dimension df < 2
Estimate of fractal dimension of percolation clusters generated by Leath method at p = 0.59
Exact answer : df = 91/48 = 1.896
p = 0.55
For p < pccorrelation length = mean distance between points on the same finite cluster
For p > pccan still define correlation length = mean distance between points on the same finite cluster.
This is typical size of holes in infinite cluster.
The infinite cluster is uniform above this length scale
p = 0.65
Minimum path length from centre.
red = shortgreen = long
almost circular contours uniform medium
p = 0.59
Minimum path length from centre.
red = shortgreen = long
irregular contours poorly connected medium
fractal
p = 0.72 Diffusion through the infinite cluster
Concentrationred = 1.0, green = 0.0
Fluxred = high, blue = low
Diffusion through the infinite cluster close to percolation
p = 0.60
Concentrationred = 1.0, green = 0.0
Fluxred = high, blue = low
Shortest path across a cluster close to pc
L = 100
L = 800
lmin ~ rdmin
dmin > 1
lmin ~ Ldmin
Shortest path across lattice of size L
Estimate in Bunde and Havlin book = 1.13
Testing the scaling hypothesis for cluster size distribution.