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Quantities of Interest in Percolation

Interactive Large Lattice

p = 0.59274621Below pc
Occupied sites are shown in grey, with the largest cluster highlighted in black. The critical probability pc ≈ 0.59274621 marks the transition point.

Key Quantities

Critical Probability (pc)

The threshold where an infinite percolating cluster first appears. For a square lattice:$p_c \approx 0.59274621$

Spanning Clusters

Clusters that connect opposite sides of the lattice. These are finite-size versions of the infinite percolating cluster.

Cluster Statistics

Current configuration has:

  • Total clusters: 0
  • Spanning clusters: 0
  • Largest cluster size: -Infinity

Key Observations:

  • When p < pc: Only finite clusters exist
  • When p = pc: "Incipient infinite clusters" emerge
  • When p > pc: A unique infinite cluster dominates
  • At pc, clusters show fractal properties