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.