Overview of Percolation Theory and Clustering Phenomena

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Explore various aspects of percolation theory, clusters, critical probabilities, and structures like the Sierpinski Gasket. Learn about site percolation, bond percolation, torus structures, and the double cover of the Sierpinski Gasket in a detailed visual format.

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  1. Percolation Clusters

  2. Site Percolation Cells are occupied with probability p, and empty with probability 1-p Clusters are groups of nearest neighboring cells

  3. For finite cases, a cluster percolates if it goes from one boundary to another p = 0.45, 0.55, 0.59, 0.65, and 0.75, respectively

  4. Bond Percolation Keep edge with probability p Remove edge with probability 1-p Cluster: network of connected components

  5. Torus Structure We are using the torus to model the lattice, Z x Z Each vertex has 4 neighbors

  6. Critical Probability Critical Probability: pcsuch that when p < pc the probability of having an infinite (or percolating) cluster is 0, and when p > pcthe probability of having an infinite cluster is 1.

  7. The Sierpinski Gasket SG level 2 Probability: 0.5

  8. SG level 4 Probability: 0.5

  9. SG level 7 Probability: 0.5

  10. Max Cluster at Level 7

  11. Double Cover of SG

  12. SG x SG (double cover)

  13. Critical Probability on SG x SG SG level 2 SG level 3

  14. SG3 x SG3and SG4 x SG4 SG level 4 SG level 3

  15. Cluster Variation SG3x SG3

  16. Cluster Variation Level 3:

  17. Number of Neighbors (SG x SG Level 4)

  18. Still To Come... Laplacian on percolation clusters Random walks Recurrent vs. Transient walks on clusters Cell graphs vs. vertex graph 3 neighbors rather than 4

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