Site percolation

1
Site percolation Square lattice 400 x 400 Three
largest clusters are coloured green/blue/yellow
p 0.50
2
p 0.55
3
p 0.58
4
p 0.59
5
p 0.60
6
p 0.65
7
Fraction of sites on the largest cluster
8
S mean size of finite clusters
Estimate in Bunde and Havlin pc 0.5927
9
Cluster generation via Leath method (epidemic
spreading)
10
space dimension d 2 fractal dimension df
M rdf
At pc the infinite cluster has a fractal
dimension df lt 2
11
Estimate of fractal dimension of percolation
clusters generated by Leath method at p 0.59
Exact answer df 91/48 1.896
12
For p lt pc correlation length ? mean distance
between points on the same finite cluster
p 0.55
13

• For p gt pc
• can 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

14
Minimum path length from centre. red
short green long almost circular contours ?
uniform medium
p 0.65
15
Minimum path length from centre. red
short green long irregular contours ? poorly
connected medium fractal
p 0.59
16
p 0.72
Diffusion through the infinite cluster
Concentration red 1.0, green 0.0
Flux red high, blue low
17
Diffusion through the infinite cluster close to
percolation
p 0.60
Concentration red 1.0, green 0.0
Flux red high, blue low
18
L 100
Shortest path across a cluster close to pc
L 800
lmin rdmin
dmin gt 1
19
Shortest path across lattice of size L
lmin Ldmin
Estimate in Bunde and Havlin book 1.13
20
Testing the scaling hypothesis for cluster size
distribution.