1. p = 0.50 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 =

9. Cluster generation via Leath method
(epidemic spreading)

10. M ~ rdf r space dimension d = 2 fractal dimension df r
At pc the infinite cluster has a fractal dimension df < 2 r

11. Estimate of fractal dimension of percolation clusters generated by Leath method at p = 0.59
Exact answer : df = 91/48 = 1.896

12. p = 0.55  For p < pc correlation length
 = mean distance between points on the same finite cluster
p = 0.55 

13.  For p > 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. p = 0.65 Minimum path length from centre. red = short green = long
almost circular contours  uniform medium
p = 0.65

15. p = 0.59 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
Concentration
red = 1.0, green = 0.0
Flux
red = high, blue = low

18. lmin ~ rdmin L = 100 Shortest path across a cluster close to pc

19. lmin ~ Ldmin Shortest path across lattice of size L
Estimate in Bunde and Havlin book = 1.13

20. Testing the scaling hypothesis for cluster size distribution.