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Site percolation Square lattice 400 x 400 Three largest clusters are coloured green/blue/yellow

Site percolation Square lattice 400 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: p c = 0.5927.

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Site percolation Square lattice 400 x 400 Three largest clusters are coloured green/blue/yellow

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  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 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. 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. 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. p = 0.60 Diffusion through the infinite cluster close to percolation 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 > 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.

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